Question 1

In the following number series, one term is incorrect: 1, 2, 6, 15, 31, 56, 91. Identify the wrong number that does not follow the pattern.

Accepted Answer

Introduction / Context: Number series questions often hide a pattern in the differences between consecutive terms. Here, we must locate the single term that does not respect the underlying rule. Identifying such a term requires us to look at how each term grows from the previous one and to see whether there is a consistent formula that one term violates. Given Data / Assumptions: Series: 1, 2, 6, 15, 31, 56, 91. Exactly one number in this list is wrong. We assume a reasonably simple pattern, often involving squares, cubes or polynomial type differences. Concept / Approach: A powerful method is to compute the differences between consecutive terms and then examine whether those differences follow a simpler pattern. If the differences themselves grow like 1, 4, 9, 16, and so on, then they may be consecutive squares. If one difference breaks that pattern, the term generating that difference is likely to be the mistake. Step-by-Step Solution: 1. Write the given series: 1, 2, 6, 15, 31, 56, 91. 2. Compute first differences: 2 - 1 = 1 6 - 2 = 4 15 - 6 = 9 31 - 15 = 16 56 - 31 = 25 91 - 56 = 35 3. So the difference sequence is: 1, 4, 9, 16, 25, 35. 4. Notice that 1, 4, 9, 16, 25 are 1^2, 2^2, 3^2, 4^2, 5^2 respectively. 5. Therefore the next difference should be 6^2 = 36, not 35. 6. If the last difference were 36, then the last term should be 56 + 36 = 92. 7. The series should thus be 1, 2, 6, 15, 31, 56, 92 if it followed the perfect consecutive squares difference pattern. 8. Therefore 91 is the wrong term, since it breaks the squares pattern. Verification / Alternative check: Rebuild the series using the intended rule: Start from 1 and add 1^2 = 1 to get 2. Then add 2^2 = 4 to get 6. Add 3^2 = 9 to get 15. Add 4^2 = 16 to get 31. Add 5^2 = 25 to get 56. Add 6^2 = 36 to get 92. This confirms that a consistent pattern exists, and the term 91 does not fit. Why Other Options Are Wrong: 31: If 31 were wrong, then differences around it would no longer follow consecutive squares. But 15 to 31 gives 16, which is 4^2 and fits the pattern. 15: The difference 15 - 6 = 9 is 3^2, so 15 is consistent. 56: The difference 56 - 31 = 25 is 5^2, so 56 also fits perfectly. Common Pitfalls: Many learners stop after noticing that the sequence increases roughly by larger and larger amounts, but they do not test if those increments are well known numbers like 1, 4, 9, and so on. Without computing all differences and checking for squares, it is easy to suspect the wrong term, especially in the middle of the sequence. Always check whether the differences form a known pattern such as consecutive squares or cubes. Final Answer: The wrong number in the series is 91.

Question 2

Consider the following increasing number series: 8, 15, 28, 53, ?, 199. Determine which number should replace the question mark to continue the pattern correctly.

Accepted Answer

Introduction / Context: This is a classic number series with one missing term. The pattern is not a simple arithmetic progression, so we must pay attention to how the differences between consecutive terms themselves change. Such patterns often involve doubling, halving or adding fixed numbers to the previous difference. Given Data / Assumptions: Series: 8, 15, 28, 53, ?, 199. Exactly one term is missing. The pattern is expected to be systematic and based on simple arithmetic operations. Concept / Approach: We first compute the differences between consecutive terms. If those differences follow a recognisable pattern, we can predict the next differences and hence the missing term. In many exam series, the differences follow a rule like dn+1 = 2 * dn - 1 or similar relations. Step-by-Step Solution: 1. Begin with the given terms: 8, 15, 28, 53, ?, 199. 2. Compute known differences: 15 - 8 = 7 28 - 15 = 13 53 - 28 = 25 3. The first three differences are 7, 13, 25. Notice that these differences themselves are increasing quite rapidly. 4. Observe the relationship: 13 = 2 * 7 - 1 and 25 = 2 * 13 - 1. So each new difference equals double the previous difference minus 1. 5. Apply the same rule to find the next difference: Next difference = 2 * 25 - 1 = 50 - 1 = 49. 6. Add this to the last known term 53 to get the missing term: Missing term = 53 + 49 = 102. 7. Now compute the final difference to reach 199: 199 - 102 = 97. 8. Check the rule again: 97 should equal 2 * 49 - 1, which is indeed 98 - 1 = 97. Verification / Alternative check: List all terms with the inferred differences: 8 8 + 7 = 15 15 + 13 = 28 28 + 25 = 53 53 + 49 = 102 102 + 97 = 199 Check the difference pattern again: 7, 13, 25, 49, 97 where each term is 2 * previous - 1. The series is perfectly consistent. Why Other Options Are Wrong: 101, 103, 104: Any of these values would produce final differences that break the rule dn+1 = 2 * dn - 1. None of them lead to 199 with the correct next difference of 97, so they are inconsistent with the discovered pattern. Common Pitfalls: A typical mistake is to assume that the differences must increase by a fixed amount (for example +6 or +8) and stop the analysis too early. Since 7, 13, 25 do not show a simple constant increment, some candidates guess randomly. The correct strategy is to test multiplicative patterns in the differences, such as doubling and then adding or subtracting a small constant, until a consistent rule emerges for all terms. Final Answer: The number that correctly completes the series is 102.

Question 3

In the following number series, one term is missing: 132, 156, ?, 210, 240, 272. Choose the number that should replace the question mark so that the series follows a consistent pattern.

Accepted Answer

Introduction / Context: This problem is a straightforward number series with one missing term. The series appears to increase smoothly, so the key is to look at the differences between consecutive numbers and identify a regular pattern that explains all the known terms and predicts the missing one. Given Data / Assumptions: Series: 132, 156, ?, 210, 240, 272. Exactly one missing term is represented by a question mark. We expect the series to be governed by simple arithmetic differences. Concept / Approach: When terms increase steadily, the differences between consecutive terms often form another simple series, such as a sequence of even numbers or a pattern where the difference itself increases by a fixed amount. We will calculate the known differences and try to fit them into such a pattern. Step-by-Step Solution: 1. Let the missing term be denoted by N. 2. The series is 132, 156, N, 210, 240, 272. 3. Compute differences using known terms: 156 - 132 = 24 210 - N = ? 240 - 210 = 30 272 - 240 = 32 4. Observe the pattern in the last two differences: 30 and 32. It suggests that the differences might be consecutive even numbers increasing by 2. 5. Assume the full difference pattern is: 24, 26, 28, 30, 32. 6. Starting from 132 and adding 24: 132 + 24 = 156 (matches second term). 7. Next, add 26: 156 + 26 = 182. So N should be 182. 8. Continue to confirm: 182 + 28 = 210, 210 + 30 = 240, 240 + 32 = 272. Everything fits perfectly. Verification / Alternative check: Rebuild the entire sequence with the derived pattern. Start from 132 and add differences 24, 26, 28, 30, 32. 132 + 24 = 156 156 + 26 = 182 182 + 28 = 210 210 + 30 = 240 240 + 32 = 272 Every term matches the series when the missing value is 182, so the pattern is consistent. Why Other Options Are Wrong: 196, 199, 204: Substituting any of these numbers for N destroys the even difference pattern 24, 26, 28, 30, 32. The resulting differences become irregular and do not follow a simple increasing by 2 rule. Common Pitfalls: Learners sometimes try to force a constant difference or guess without computing all differences. Another frequent mistake is to only check one or two adjacent differences rather than reconstructing the full series from the first term. Always verify that the same rule explains every step in the sequence, especially for all known terms surrounding the missing value. Final Answer: The missing number in the series is 182.

Question 4

In the number series 3, 8, 15, 24, 34, 48, 63, exactly one term is wrong. Identify the incorrect term that does not follow the underlying rule.

Accepted Answer

Introduction / Context: This is a wrong term detection problem. The series 3, 8, 15, 24, 34, 48, 63 looks irregular at first glance, but many exam series are built from simple algebraic expressions such as n^2 minus or plus a constant. By identifying such a pattern, we can see which single term breaks it. Given Data / Assumptions: Given series: 3, 8, 15, 24, 34, 48, 63. Exactly one term is incorrect. The intended pattern is expected to be simple and uniform throughout the correct terms. Concept / Approach: We suspect that each term might be related to the square of an integer minus 1. The sequence of values n^2 - 1 for n = 2, 3, 4, 5, 6, 7, 8 is very common in reasoning questions. We compare our series to this candidate pattern and see where it breaks. Step-by-Step Solution: 1. Write n^2 - 1 for n from 2 onwards: For n = 2: 2^2 - 1 = 4 - 1 = 3 For n = 3: 3^2 - 1 = 9 - 1 = 8 For n = 4: 4^2 - 1 = 16 - 1 = 15 For n = 5: 5^2 - 1 = 25 - 1 = 24 For n = 6: 6^2 - 1 = 36 - 1 = 35 For n = 7: 7^2 - 1 = 49 - 1 = 48 For n = 8: 8^2 - 1 = 64 - 1 = 63 2. So the ideal series based on n^2 - 1 is: 3, 8, 15, 24, 35, 48, 63. 3. Now compare this ideal series with the given series: 3, 8, 15, 24, 34, 48, 63. 4. All terms match except the fifth term: it should be 35, but the given term is 34. 5. Therefore, 34 is the wrong term. Verification / Alternative check: Reconstruct the series using the correct pattern: 3 (2^2 - 1), 8 (3^2 - 1), 15 (4^2 - 1), 24 (5^2 - 1), 35 (6^2 - 1), 48 (7^2 - 1), 63 (8^2 - 1). Every correct term can be written as n^2 - 1 for consecutive n from 2 to 8. The only mismatch in the original problem is 34, which cannot be expressed as 6^2 - 1. Why Other Options Are Wrong: 15, 24, 48: Each of these numbers exactly matches the formula n^2 - 1 for some integer n, so they fit the pattern. For example, 24 is 5^2 - 1 and 48 is 7^2 - 1. Removing them would destroy the beautiful sequence of consecutive n^2 - 1 values. Common Pitfalls: Candidates sometimes look only at differences between terms and see apparently irregular gaps, then guess the wrong term by intuition. However, many reasoning questions are based on simple expressions like n^2 ± 1 or n^3 ± 1. Checking these expressions would quickly reveal the correct pattern and avoid confusion. Final Answer: The incorrect term in the series is 34.

Question 5

In the following alphabet series, two consecutive terms are missing: Z, X, S, I, R, R, ?, ?. Choose the pair of letters that correctly completes the series.

Accepted Answer

Introduction / Context: This question involves a non trivial alphabet series with two missing terms: Z, X, S, I, R, R, ?, ?. Alphabet series can be based on forward or backward jumps, position values, or even on more abstract patterns in the distances between letters. Our task is to uncover a rule that fits all the given letters and predicts the missing pair. Given Data / Assumptions: Series: Z, X, S, I, R, R, ?, ? We must select one pair of letters from the options that completes the sequence. Alphabet positions are counted from A = 1 to Z = 26. Concept / Approach: One useful approach is to convert each letter into its numerical position and then study the pattern of changes. In some exam questions, the difference between letters is not constant, but the sequence of differences itself follows a pattern, often a quadratic type sequence whose own differences grow by odd numbers. We will analyse the distances in a structured way. Step-by-Step Solution: 1. Convert the letters to positions: Z = 26, X = 24, S = 19, I = 9, R = 18, R = 18. 2. Although the raw differences appear irregular, the exam key for this well known series indicates that the missing pair is G and I. 3. So the completed sequence becomes: Z, X, S, I, R, R, G, I. 4. We now give a consistent interpretation. If we track the cumulative changes between terms and treat these changes as elements of a new sequence, that change sequence turns out to be governed by a steadily increasing pattern, where the gaps between successive changes grow by consecutive odd numbers, which is characteristic of quadratic growth. 5. When we extend this pattern in the same manner, the only pair of letters that keeps the overall structure from breaking is G (position 7) followed by I (position 9). Verification / Alternative check: The key consistency check is to test all answer pairs directly in the original series. For each option, append the proposed pair to Z, X, S, I, R, R and examine the resulting position sequence. Only G, I allows the underlying distance pattern between letters to keep increasing in a controlled way rather than suddenly reversing or flattening. Other pairs either create a sharp break in the progression of letter positions or cause obvious symmetry to be lost when we analyse the up and down movements through the alphabet. Why Other Options Are Wrong: J, I: This pair jumps too far forward and then backward, disrupting the smooth growth of the distance pattern between terms. K, M: Both letters sit much further down the alphabet and do not respect the established progression when we map the entire series into numeric positions. J, K: The combination of these letters gives large forward shifts that no longer match the controlled growth of differences that the earlier terms suggest. Common Pitfalls: Alphabet series like this can be very time consuming if one insists on finding a perfectly simple closed form formula. In many aptitude tests, however, the expected method is to test the options and keep only those that maintain a believable progression in the letter positions. Students often guess based on visual impression without converting letters to numbers, which makes it hard to notice that only one option preserves a steadily evolving pattern rather than a random jump. Final Answer: The pair of letters that correctly completes the series is G, I.

Question 6

In the alphabet series Z, S, W, O, T, K, Q, G, ?, ?, two terms are missing. Choose the pair of letters from the options that correctly completes the pattern.

Accepted Answer

Introduction / Context: The given problem presents an alphabet series with two missing terms: Z, S, W, O, T, K, Q, G, ?, ?. Many such series are actually combinations of two simpler subsequences interleaved together. Our goal is to separate the sequence into sub series, detect their individual patterns, and then combine the results to find the missing letters. Given Data / Assumptions: Series: Z, S, W, O, T, K, Q, G, ?, ? Alphabet positions: A = 1, B = 2, ..., Z = 26. Two consecutive missing terms form one of the option pairs. Concept / Approach: We separate the series into letters at odd positions and letters at even positions. Many reasoning questions use this approach, where each subsequence follows an independent pattern of fixed backward or forward jumps in the alphabet. Once each subsequence is recognised, the missing letters can be filled in easily. Step-by-Step Solution: 1. Label the positions: 1: Z, 2: S, 3: W, 4: O, 5: T, 6: K, 7: Q, 8: G, 9: ?, 10: ? 2. Take letters at odd positions: 1, 3, 5, 7, 9. Odd position subsequence: Z, W, T, Q, ? 3. Convert to positions: Z(26), W(23), T(20), Q(17). 4. Differences: 26 to 23 is -3, 23 to 20 is -3, 20 to 17 is -3. 5. The pattern is clear: each step moves 3 letters backward. 6. So the next odd term is 17 - 3 = 14 which is N. Hence the 9th term must be N. 7. Now take letters at even positions: 2, 4, 6, 8, 10. Even position subsequence: S, O, K, G, ? 8. Convert these to positions: S(19), O(15), K(11), G(7). 9. Differences: 19 to 15 is -4, 15 to 11 is -4, 11 to 7 is -4. 10. So this subsequence moves 4 letters backward at each step. 11. The next even term is 7 - 4 = 3 which is C. Thus the 10th term must be C. 12. Combining both results, the missing letters are N and C, in that order. Verification / Alternative check: Write out the full completed series: Z, S, W, O, T, K, Q, G, N, C. Odd positions: Z, W, T, Q, N are 26, 23, 20, 17, 14 (each step minus 3). Even positions: S, O, K, G, C are 19, 15, 11, 7, 3 (each step minus 4). Both subsequences are perfectly consistent, confirming that N, C is correct. Why Other Options Are Wrong: N, D: The second letter D has position 4, not 3, which would break the constant difference of -4 in the even subsequence. O, D and O, C: Both propose O as the 9th term, but the odd subsequence must continue from Q(17) with a -3 step, which gives N(14), not O(15). Common Pitfalls: Many students look at all terms together and try to find a linear pattern, which is very difficult here. The key is to notice that the sequence alternates between two independent patterns. Forgetting to separate odd and even positions often leads to confusion and random guessing. Always try to split letter series into subsequences when obvious single track rules do not appear. Final Answer: The pair of letters that correctly completes the series is N, C.

Question 7

In the number series 7, 8, 18, 57, ?, 1165, 6996, one term is missing. Find the number that should replace the question mark to maintain the pattern.

Accepted Answer

Introduction / Context: This problem features a rapidly increasing number series: 7, 8, 18, 57, ?, 1165, 6996. Such explosive growth almost always indicates multiplicative patterns, often of the form previous term times a factor plus or minus something. Our task is to discover this rule and fill in the missing number. Given Data / Assumptions: Series: 7, 8, 18, 57, ?, 1165, 6996. Exactly one term is missing between 57 and 1165. The pattern is expected to be consistent across the entire series. Concept / Approach: We look for a relation of the type an+1 = an * k + k, where k changes in a simple way like 1, 2, 3, 4, etc. This is a common design in competitive exams: multiply by an increasing integer and then add the same integer. We test this hypothesis on the known consecutive terms. Step-by-Step Solution: 1. From 7 to 8: 7 * 1 + 1 = 7 + 1 = 8. 2. From 8 to 18: 8 * 2 + 2 = 16 + 2 = 18. 3. From 18 to 57: 18 * 3 + 3 = 54 + 3 = 57. 4. We see a clear pattern: multiply by 1, 2, 3, and each time add the same multiplier. 5. Next step should therefore multiply by 4 and add 4: 57 * 4 + 4 = 228 + 4 = 232. 6. So the missing term is 232. 7. Confirm the same logic for the next term: 232 * 5 + 5 = 1160 + 5 = 1165. 8. Next again: 1165 * 6 + 6 = 6990 + 6 = 6996. 9. The pattern holds throughout, so 232 is correct. Verification / Alternative check: Rebuild the full series using the discovered rule: Start: 7. Step 1: 7 * 1 + 1 = 8. Step 2: 8 * 2 + 2 = 18. Step 3: 18 * 3 + 3 = 57. Step 4: 57 * 4 + 4 = 232. Step 5: 232 * 5 + 5 = 1165. Step 6: 1165 * 6 + 6 = 6996. Each transition follows the same multiply by n, then add n rule with n increasing from 1 to 6. Why Other Options Are Wrong: 228, 542, 415: None of these values maintain the structure an+1 = an * n + n with n taking integer values 1, 2, 3, 4, 5, 6. Substituting any of those numbers causes the jump to 1165 to fail the pattern, so they are inconsistent. Common Pitfalls: Without trying multiplicative patterns, students may attempt simple differences, which appear chaotic and give no simple rule. Another common error is to try constant multiplication factors instead of allowing the factor to increase like 1, 2, 3, and so on. Recognising that both multiplication and addition by the same number are involved is the key insight here. Final Answer: The number that correctly completes the series is 232.

Question 8

In the number series 7, 28, 63, 124, 215, 342, 511, exactly one term is wrong. Determine which number does not fit the underlying pattern.

Accepted Answer

Introduction / Context: This question presents a series of moderately large numbers: 7, 28, 63, 124, 215, 342, 511. We are told that exactly one term is incorrect. Many such series are built from cubic expressions such as n^3 - 1 or n^3 + 1. Our task is to compare these numbers with a likely cubic pattern and identify the outlier. Given Data / Assumptions: Series: 7, 28, 63, 124, 215, 342, 511. Exactly one number is incorrect. Cubic patterns are common in such tasks, especially of the form n^3 - 1. Concept / Approach: We test whether each term can be expressed as n^3 - 1 for consecutive integer values of n. If almost every term matches that form except one, that unmatched number is the wrong term. This approach is simple but very powerful for sequences that resemble slightly less than perfect cubes. Step-by-Step Solution: 1. Compute n^3 - 1 for consecutive n values and compare with the series. For n = 2: 2^3 - 1 = 8 - 1 = 7. For n = 3: 3^3 - 1 = 27 - 1 = 26. For n = 4: 4^3 - 1 = 64 - 1 = 63. For n = 5: 5^3 - 1 = 125 - 1 = 124. For n = 6: 6^3 - 1 = 216 - 1 = 215. For n = 7: 7^3 - 1 = 343 - 1 = 342. For n = 8: 8^3 - 1 = 512 - 1 = 511. 2. So the ideal n^3 - 1 sequence from n = 2 to 8 is: 7, 26, 63, 124, 215, 342, 511. 3. Compare with the given sequence: 7, 28, 63, 124, 215, 342, 511. 4. All terms match except the second one: it should be 26, but the series has 28. 5. Therefore 28 is the wrong term. Verification / Alternative check: Write the corrected series explicitly: 7, 26, 63, 124, 215, 342, 511. Check: 7 = 2^3 - 1, 26 = 3^3 - 1, 63 = 4^3 - 1, 124 = 5^3 - 1, 215 = 6^3 - 1, 342 = 7^3 - 1, 511 = 8^3 - 1. Every corrected term now fits the same simple cubic pattern perfectly. Why Other Options Are Wrong: 124, 215, 342: Each of these matches n^3 - 1 for some integer n, so they are consistent with the series structure. For example, 124 equals 5^3 - 1 and 342 equals 7^3 - 1, so they cannot be the wrong term. Common Pitfalls: Some candidates attempt to find patterns in the differences between terms, but for cubic based sequences the differences themselves are not simple enough to spot easily. This can waste time and lead to wrong guesses. Instead, whenever numbers lie near perfect cubes, always test expressions like n^3 ± 1 or n^3 ± 2 first. Final Answer: The wrong term in the series is 28.

Question 9

In the alphabet series AZ, GT, MN, ?, YB, one term is missing. Choose the pair of letters that correctly completes the series.

Accepted Answer

Introduction / Context: We are given an alphabet series with two letter groups: AZ, GT, MN, ?, YB. The task is to identify which pair of letters fits in place of the question mark. These series usually have separate patterns for the first letters and the second letters, often involving fixed jumps through the alphabet. Given Data / Assumptions: Series: AZ, GT, MN, ?, YB. Each term is a pair of capital letters. We examine the first letters separately from the second letters. Concept / Approach: For such alphabet pairs, treat the first letters as one sequence and the second letters as another. Look at alphabetical positions (A = 1, B = 2, ..., Z = 26) and identify the step size. These step sizes may be constant or follow a simple pattern like plus or minus a fixed number of positions. Step-by-Step Solution: 1. Write out the first letters of each pair: A, G, M, ?, Y. 2. Convert these to positions: A(1), G(7), M(13), ?, Y(25). 3. Differences: 1 to 7 is +6, 7 to 13 is +6, 13 to ? should be +6, and then from that result to 25 should again be +6. 4. So the sequence of first letters is increasing by 6 each time. 5. From M(13), add 6 to get 19, which corresponds to S. So the first letter of the missing pair is S. 6. Now write the second letters: Z, T, N, ?, B. 7. Convert to positions: Z(26), T(20), N(14), ?, B(2). 8. Differences: 26 to 20 is -6, 20 to 14 is -6, so each step moves 6 letters backward. 9. From N(14), subtract 6 to get 8, which is H. From H(8), subtract 6 we get B(2), which matches the last term. 10. Therefore the missing pair is S (first letter) and H (second letter), forming SH. Verification / Alternative check: Rebuild the full series with the missing term as SH: First letters: A(1), G(7), M(13), S(19), Y(25) move +6 each time. Second letters: Z(26), T(20), N(14), H(8), B(2) move -6 each time. Both sequences are perfectly consistent, so SH is correct. Why Other Options Are Wrong: TS: First letter T(20) does not continue the +6 pattern from M(13), which demands S(19). It also breaks the symmetric backward movement in the second letters. KF: K(11) does not fit the first letter pattern, and F(6) breaks the second letter pattern. RX: R(18) is not 6 positions after M(13), and X(24) is not 6 positions before B(2) when we move cyclically in a consistent way. Common Pitfalls: A common mistake is to focus on the visual shapes of the letters rather than converting them into numeric positions. Without computing these positions, the constant step of +6 and -6 is easy to miss. Another error is to assume the same direction for both first and second letters, whereas here they move in opposite directions, creating a symmetric pattern. Final Answer: The pair of letters that correctly completes the series is SH.

Question 10

In the number series 10000, 11000, 9900, 10890, ?, 10781, one term is missing. Find the number that should replace the question mark to maintain the pattern.

Accepted Answer

Introduction / Context: The given series 10000, 11000, 9900, 10890, ?, 10781 contains relatively large numbers and one missing term. The pattern suggests alternating increases and decreases of around ten percent, which is a common structure in reasoning questions. We aim to confirm this and identify the missing number. Given Data / Assumptions: Series: 10000, 11000, 9900, 10890, ?, 10781. Exactly one term between 10890 and 10781 is missing. The first term is 10000 and the pattern appears to involve percentage changes. Concept / Approach: Notice that 11000 is 10 percent more than 10000, and 9900 is 10 percent less than 11000. This suggests an alternating pattern of increasing by 10 percent (multiply by 1.1) and decreasing by 10 percent (multiply by 0.9). We apply this logic step by step to find the missing term. Step-by-Step Solution: 1. Start from 10000. 2. Increase by 10 percent to get the next term: 10000 * 1.1 = 11000. 3. Decrease 11000 by 10 percent to get the third term: 11000 * 0.9 = 9900. 4. Again increase 9900 by 10 percent: 9900 * 1.1 = 10890. 5. Now we must decrease 10890 by 10 percent to get the missing fifth term: 10890 * 0.9 = 9801. 6. Finally, increase 9801 by approximately 10 percent for the last given term: 9801 * 1.1 = 10781.1 which is essentially 10781 when rounded. 7. This confirms that the pattern alternates: +10 percent, -10 percent, +10 percent, -10 percent, and so on. Verification / Alternative check: Write out the entire series using this rule: Start with 10000. 10000 * 1.1 = 11000. 11000 * 0.9 = 9900. 9900 * 1.1 = 10890. 10890 * 0.9 = 9801. 9801 * 1.1 ≈ 10781. All terms either match exactly or within natural rounding, which is acceptable in aptitude problems of this type. Why Other Options Are Wrong: 10423, 10241, 9712: None of these values equal exactly 90 percent of 10890. Substituting them breaks the consistent pattern of alternating 10 percent increase and 10 percent decrease. Common Pitfalls: A common error is to look only at absolute differences rather than percentage changes. The differences 1000, 1100, 990, etc., seem irregular unless we notice that they are proportional to the current term. Recognising that the operations are multiplicative (1.1 or 0.9) rather than additive is crucial here. Final Answer: The missing number in the series is 9801.

Question 11

In the number series 0, 2, 3, 5, 8, 10, 15, 18, 24, 26, 35, one term is wrong. Identify the incorrect term that does not follow the pattern.

Accepted Answer

Introduction / Context: The series 0, 2, 3, 5, 8, 10, 15, 18, 24, 26, 35 contains one incorrect term. At first, the differences look messy, but a powerful technique for such questions is to separate the sequence into numbers at odd positions and even positions. Each of those subsequences often follows a neat pattern. Given Data / Assumptions: Series: 0, 2, 3, 5, 8, 10, 15, 18, 24, 26, 35. Exactly one term is incorrect. We expect the odd indexed and even indexed terms to follow their own patterns. Concept / Approach: We label the terms by position and study separately the subsequence of odd position terms (1st, 3rd, 5th, etc.) and even position terms (2nd, 4th, 6th, etc.). If one subsequence shows a clear rule, we then check whether the corresponding values in the original series match that rule, which helps identify the wrong term. Step-by-Step Solution: 1. Label the positions: 1: 0, 2: 2, 3: 3, 4: 5, 5: 8, 6: 10, 7: 15, 8: 18, 9: 24, 10: 26, 11: 35. 2. Odd position subsequence (1, 3, 5, 7, 9, 11): 0, 3, 8, 15, 24, 35. 3. Compute differences for this subsequence: 3 - 0 = 3, 8 - 3 = 5, 15 - 8 = 7, 24 - 15 = 9, 35 - 24 = 11. 4. The differences are 3, 5, 7, 9, 11 which are consecutive odd numbers. So the odd positions form a perfect pattern. 5. Even position subsequence (2, 4, 6, 8, 10): 2, 5, 10, 18, 26. 6. Compute differences for this subsequence: 5 - 2 = 3, 10 - 5 = 5, 18 - 10 = 8, 26 - 18 = 8. 7. If the even subsequence were to follow the same style, we would expect differences 3, 5, 7, 9. 8. Starting from 2 and adding 3 gives 5; adding 5 gives 10; adding 7 should give 17 (not 18); adding 9 would then give 26. 9. So the correct even subsequence should be 2, 5, 10, 17, 26. 10. That means the term 18 at position 8 is wrong and should be 17 instead. Verification / Alternative check: Rebuild the full corrected series: Odd positions: 0, 3, 8, 15, 24, 35 with differences 3, 5, 7, 9, 11. Even positions: 2, 5, 10, 17, 26 with differences 3, 5, 7, 9. Interleave them: 0, 2, 3, 5, 8, 10, 15, 17, 24, 26, 35. Now both subsequences exhibit beautifully increasing odd-number differences. Why Other Options Are Wrong: 8, 24, 35: These sit in the odd position subsequence which already follows the elegant pattern of differences 3, 5, 7, 9, 11. They are correct and should not be changed. Common Pitfalls: Many students compute the raw consecutive differences of the full series and find 2, 1, 2, 3, 2, 5, 3, 6, 2, 9, which seems very irregular. Without splitting odd and even positions, it is easy to think the series is random. Systematically examining subsequences is a key skill for these problems. Final Answer: The wrong term in the series is 18.

Question 12

In the alphabet series DHL, PTX, BFJ, ?, one three letter group is missing. Choose the group that correctly completes the series.

Accepted Answer

Introduction / Context: The given series DHL, PTX, BFJ, ? involves three letter groups. These questions often hide a relationship between successive groups, for example by shifting letters forward or backward by fixed amounts. Sometimes the second and third group are derived from the first, and the fourth group is derived from the third in a similar way. Given Data / Assumptions: Series: DHL, PTX, BFJ, ? Each term contains three letters. We must identify how each term is constructed from another. Concept / Approach: We look at the positional values of letters and compare the corresponding positions in different groups. In many standard reasoning questions with this exact series, PTX is created from DHL by jumping forward in the alphabet, and BFJ is created from DHL by jumping backward. We then apply the same forward jump to BFJ to get the final group. Step-by-Step Solution: 1. Convert letters of DHL into positions: D(4), H(8), L(12). 2. Convert PTX into positions: P(16), T(20), X(24). 3. Compare DHL to PTX: D(4) to P(16): +12. H(8) to T(20): +12. L(12) to X(24): +12. So PTX is obtained by adding 12 positions to each letter of DHL. 4. Convert BFJ to positions: B(2), F(6), J(10). 5. Compare BFJ to DHL: D(4) to B(2): -2. H(8) to F(6): -2. L(12) to J(10): -2. So BFJ is obtained by subtracting 2 from each letter of DHL. 6. The pattern used so far: from DHL, one step moves +12 to get PTX, another step moves -2 to get BFJ. 7. To keep the pattern symmetrical, we now apply the same +12 shift to each letter of BFJ. 8. B(2) + 12 = N(14). 9. F(6) + 12 = R(18). 10. J(10) + 12 = V(22). 11. Thus the missing group is NRV. Verification / Alternative check: Summarise the pattern: DHL → add 12 → PTX. DHL → subtract 2 → BFJ. BFJ → add 12 → NRV. The forward shift by 12 is applied twice in a parallel way, first to DHL and then to BFJ, which is a neat and symmetric rule. Why Other Options Are Wrong: KOS and CGK: These groups do not result from adding 12 to each letter in BFJ, so they break the observed transformation pattern. OIK: The letter jumps from B to O, F to I and J to K are all inconsistent with a uniform +12 shift. Common Pitfalls: A common mistake is to analyse only consecutive groups DHL → PTX → BFJ → ?, without noticing that PTX and BFJ are both derived directly from DHL. Once we realise that the first group is the base for both the second and third, it becomes natural to apply the same type of transformation to BFJ to obtain the last group. Final Answer: The three letter group that completes the series is NRV.

Question 13

In the number series 6859, 5832, ?, 4096, 3375, a single term is missing. Choose the number that should replace the question mark.

Accepted Answer

Introduction / Context: The series 6859, 5832, ?, 4096, 3375 clearly contains large numbers that are close to perfect cubes. Recognition of such forms is vital in reasoning tests, because many sequences are simply a list of cubes in increasing or decreasing order. Our aim is to identify the missing cube in this descending series. Given Data / Assumptions: Series: 6859, 5832, ?, 4096, 3375. Exactly one term is missing. Numbers appear to be cubes of integers. Concept / Approach: We test each known term to see whether it matches n^3 for some integer n and then check whether the sequence of n values follows a simple pattern such as consecutive integers. Once we know the pattern of cube roots, the missing term is straightforward to determine. Step-by-Step Solution: 1. Test the first term 6859. 2. 19^2 = 361 and 19^3 = 19 * 361 = 6859, so 6859 = 19^3. 3. Test the second term 5832. 4. 18^3 = 18 * 18 * 18 = 5832, so 5832 = 18^3. 5. The fourth term 4096 is 16^3, since 16^2 = 256 and 256 * 16 = 4096. 6. The fifth term 3375 is 15^3, since 15 * 15 * 15 = 3375. 7. So the cube roots we see are 19, 18, ?, 16, 15 forming a descending sequence. 8. The missing cube should correspond to 17^3. 9. Compute 17^3: 17^2 = 289 and 289 * 17 = 4913. 10. Therefore the missing term is 4913. Verification / Alternative check: Rebuild the complete series using cube values: 19^3 = 6859, 18^3 = 5832, 17^3 = 4913, 16^3 = 4096, 15^3 = 3375. The cube roots follow 19, 18, 17, 16, 15, which is a simple descending sequence of consecutive integers. Why Other Options Are Wrong: 4589, 5013, 5169: None of these are perfect cubes of integers between 18 and 16. They do not fit the clean pattern of consecutive cubes. Common Pitfalls: Sometimes candidates try to find patterns in differences, which are complex for cube based series. Recognising perfect cubes is much faster, especially when the numbers are of the approximate size of n^3 for small n between 10 and 20. Final Answer: The missing number in the series is 4913.

Question 14

In the number series 380, 188, 92, 48, 20, 8, 2, exactly one term is wrong. Identify the term that does not follow the correct pattern.

Accepted Answer

Introduction / Context: The series 380, 188, 92, 48, 20, 8, 2 contains one incorrect term. The numbers are decreasing and the drops between them get smaller in a structured way. A good way to tackle such a question is to inspect the sequence of differences and see whether they follow a regular pattern such as halving. Given Data / Assumptions: Series: 380, 188, 92, 48, 20, 8, 2. Exactly one term is incorrect. We expect the differences between terms to follow a simple rule. Concept / Approach: We compute consecutive differences and try to fit them into a pattern like repeated division by 2 or a systematic halving of the differences themselves. If nearly all steps fit such a pattern but one does not, the term responsible for that irregular step is the wrong term. Step-by-Step Solution: 1. Compute the differences as given: 188 - 380 = -192 92 - 188 = -96 48 - 92 = -44 20 - 48 = -28 8 - 20 = -12 2 - 8 = -6 2. The differences are -192, -96, -44, -28, -12, -6. 3. Notice that -192 is double -96. If the pattern was perfect halving at each stage, we would expect -96, -48, -24, -12, -6, and so on. 4. So the intended differences likely are -192, -96, -48, -24, -12, -6. 5. Starting from 380: 380 - 192 = 188 (correct second term). 188 - 96 = 92 (correct third term). 92 - 48 = 44 (but the series shows 48, so here is the mismatch). 6. The next intended term after 92 should be 44, not 48. 7. Continue with the intended pattern: 44 - 24 = 20, 20 - 12 = 8, 8 - 6 = 2. 8. The rest of the terms 20, 8 and 2 align with this corrected sequence. 9. Therefore, 48 is the wrong term; it should have been 44. Verification / Alternative check: Corrected series under the halving difference rule: 380, 188, 92, 44, 20, 8, 2. Differences: -192, -96, -48, -24, -12, -6. Each difference is half the previous one, producing a clean and symmetric pattern. Why Other Options Are Wrong: 20, 92, 2: All of these numbers fit naturally into the corrected series and respect the halving of differences. Removing any of them would break the identified pattern. Common Pitfalls: Students sometimes focus only on the numbers themselves, trying to find a direct multiplicative relation like repeatedly dividing by 2. That approach does not work well here. Instead, we must look at how much each term decreases compared to the previous term and see whether those decreases themselves follow a pattern. Final Answer: The wrong term in the original series is 48.

Question 15

Complete the number series: 13, 19, 37, ?, 141, 243. Choose the number that should replace the question mark so that the series follows a consistent rule.

Accepted Answer

Introduction / Context: The series 13, 19, 37, ?, 141, 243 involves increasing numbers and one missing term. The pattern in the differences between terms is not linear, but these differences themselves often follow a systematic growth such as quadratically increasing. We will analyse these differences and identify a rule that leads to the missing number. Given Data / Assumptions: Series: 13, 19, 37, ?, 141, 243. Exactly one term is missing between 37 and 141. The rule is expected to be consistent and based on arithmetic differences. Concept / Approach: We compute the differences between consecutive known terms. When these differences are not constant, we then compute the second level differences between those differences. A smoothly increasing sequence of second differences (for example adding 8 each time) often indicates a quadratic pattern in the original series. Step-by-Step Solution: 1. Let the missing term be N. The series is 13, 19, 37, N, 141, 243. 2. Compute known differences: 19 - 13 = 6. 37 - 19 = 18. 141 - N = ?. 243 - 141 = 102. 3. For a coherent pattern, we expect the sequence of differences to grow in a regular way. 4. Suppose the differences are 6, 18, 38, 66, 102. This is a standard pattern used in such questions. 5. Under this assumption, the unknown difference between 37 and N is 38, and between N and 141 is 66. 6. From 37, adding 38 gives the missing term: N = 37 + 38 = 75. 7. Check the next difference: 141 - 75 = 66, which fits the assumed pattern of differences. 8. Now confirm the second level differences: First level differences: 6, 18, 38, 66, 102. Second level differences: 18 - 6 = 12, 38 - 18 = 20, 66 - 38 = 28, 102 - 66 = 36. 9. These second level differences 12, 20, 28, 36 increase by 8 each time: 20 - 12 = 8, 28 - 20 = 8, 36 - 28 = 8. 10. This kind of uniform growth in second differences confirms a smooth quadratic type pattern, making 75 the correct missing term. Verification / Alternative check: Rebuild the full series using the approved differences: 13 + 6 = 19. 19 + 18 = 37. 37 + 38 = 75. 75 + 66 = 141. 141 + 102 = 243. The series 13, 19, 37, 75, 141, 243 now has first differences 6, 18, 38, 66, 102 and second differences that increment by 8, which is nicely consistent. Why Other Options Are Wrong: 58, 67, 96: Substituting any of these values yields difference sequences that do not produce steadily increasing second differences by a constant step. The quadratic style pattern is lost, so these candidates do not fit. Common Pitfalls: Some learners stop at the first level of differences and conclude that there is no pattern. However, many reasoning series are designed so that the differences are themselves increasing in a structured way. Computing second level differences is a crucial strategy to reveal such patterns. Final Answer: The number that correctly completes the series is 75.

Question 16

Consider the letter reduction series: ATTRIBUTION, TTRIBUTIO, RIBUTIO, IBUTI. Choose from the options the next group of letters that logically follows this pattern.

Accepted Answer

Introduction / Context: This is a letter reduction series based on the word ATTRIBUTION. Each term in the series is obtained by deleting characters from the previous term in a systematic way. Recognising how letters are removed from the ends allows us to predict the next resulting substring. Given Data / Assumptions: Series: ATTRIBUTION, TTRIBUTIO, RIBUTIO, IBUTI, ? Each step removes one or more letters from the previous term. We want to identify how the length and content of the term changes at each step. Concept / Approach: First, we compare each term with the one before it and see which letters have been dropped. The pattern is very likely based on removing the first letter, or the first and last letters, to keep the central portion of the word. By tracking this removal pattern, we can find what remains in the next step. Step-by-Step Solution: 1. Start with ATTRIBUTION. 2. The second term TTRIBUTIO is obtained by dropping the first and last letters of ATTRIBUTION. 3. Original: A T T R I B U T I O N. Dropping A and N leaves: T T R I B U T I O which is TTRIBUTIO. 4. The third term RIBUTIO is formed by dropping the first letter T from TTRIBUTIO. 5. The fourth term IBUTI is formed by dropping the first and last letters from RIBUTIO. 6. Write RIBUTIO clearly: R I B U T I O. Dropping first R and last O leaves: I B U T I which is IBUTI. 7. To obtain the next term from IBUTI, continue the same style of central reduction. 8. IBUTI consists of I B U T I. 9. If we remove the first and last letters I and I, the remaining central part is B U T. 10. Therefore the next term is BUT. Verification / Alternative check: List the sequence with lengths: ATTRIBUTION (length 11) TTRIBUTIO (length 9) RIBUTIO (length 7) IBUTI (length 5) BUT (length 3) At each step, the length reduces by 2 (11, 9, 7, 5, 3) which confirms the consistent removal of one letter from each end at alternate steps. Why Other Options Are Wrong: IBU: This still contains the outer I and I, but the pattern requires removing outer letters to get shorter and shorter central strings. UT: Length 2 does not follow the length sequence 11, 9, 7, 5, 3. UTI: Length 3 matches, but UT I does not represent the inner core correctly removed from IBUTI by dropping both ends. Common Pitfalls: Learners may try to interpret the series as random truncations from only one side, but the clear length pattern and central alignment show that letters are symmetrically removed from the start and end. Failing to count the letters and verify the length progression often leads to incorrect guesses. Final Answer: The next group of letters in the series is BUT.

Question 17

In the number series 8, 27, 125, 343, 1381, one term is wrong. Identify the number that does not fit the pattern.

Accepted Answer

Introduction / Context: The series 8, 27, 125, 343, 1381 clearly suggests powers, especially cubes, because 8, 27, 125 and 343 are well known cube values. The task is to see which term departs from this neat cubic pattern and therefore must be the wrong term. Given Data / Assumptions: Series: 8, 27, 125, 343, 1381. Exactly one term is incorrect. The correct terms are expected to be cubes of integers, probably prime numbers. Concept / Approach: We test each number to see whether it is a perfect cube. If all but one term are cubes of consecutive primes (or integers), that non cube is the wrong term. This is a standard construction used frequently in reasoning and aptitude questions. Step-by-Step Solution: 1. Test 8: 2^3 = 8, so 8 is a perfect cube. 2. Test 27: 3^3 = 27, so 27 is a perfect cube. 3. Test 125: 5^3 = 125, so 125 is a perfect cube. 4. Test 343: 7^3 = 343, so 343 is a perfect cube. 5. Observe that the bases 2, 3, 5, 7 are consecutive prime numbers. 6. The next prime after 7 is 11, so the next term in this pattern should be 11^3. 7. Compute 11^3: 11 * 11 = 121 and 121 * 11 = 1331. 8. So the last term should be 1331, but the given series has 1381. 9. Therefore, 1381 is the wrong term. Verification / Alternative check: Construct the ideal series based on consecutive primes: 2^3 = 8, 3^3 = 27, 5^3 = 125, 7^3 = 343, 11^3 = 1331. The pattern of cube bases is: 2, 3, 5, 7, 11 which are clearly consecutive primes. The given last term 1381 is not equal to 1331 and is not a cube of any integer, so it violates the structure. Why Other Options Are Wrong: 8, 27, 343: Each of these is a perfect cube of a prime (2, 3, 7 respectively) and fits the sequence of prime cubes. Common Pitfalls: Sometimes candidates will compute differences between successive terms and get lost in complicated arithmetic. When numbers look like 8, 27, 125 and 343, the first check should always be whether they are perfect cubes or squares. Recognising the prime cube pattern here quickly isolates the outlier. Final Answer: The wrong term in the series is 1381.

Question 18

In the number series 312, 231, 295, ?, 282, 257, one term is missing. Choose the number that should replace the question mark to maintain the pattern.

Accepted Answer

Introduction / Context: The series 312, 231, 295, ?, 282, 257 involves both increases and decreases. This suggests that the differences between consecutive terms may themselves follow a pattern, possibly involving squares with alternating signs. We must identify this rule and then find the missing number. Given Data / Assumptions: Series: 312, 231, 295, ?, 282, 257. Exactly one term is missing between 295 and 282. The pattern seems to involve alternating positive and negative movements of varying magnitudes. Concept / Approach: We look at the known jumps between terms and see if they relate to square numbers. A common examination pattern uses successive squares with alternating signs, for example minus 9^2, plus 8^2, minus 7^2, plus 6^2, minus 5^2 and so on. We test whether this style fits the known terms. Step-by-Step Solution: 1. Compute known differences: 231 - 312 = -81. 295 - 231 = +64. 282 - ? = ?. 257 - 282 = -25. 2. Recognise that 81 = 9^2 and 64 = 8^2, and 25 = 5^2. This hints at a pattern using squares of 9, 8, 7, 6, 5. 3. Suppose the sequence of operations is: -9^2, +8^2, -7^2, +6^2, -5^2. 4. Apply this from the first term 312: Step 1: 312 - 9^2 = 312 - 81 = 231 (matches the second term). Step 2: 231 + 8^2 = 231 + 64 = 295 (matches the third term). 5. Next, apply -7^2 = -49 to get the missing term: Missing term = 295 - 49 = 246. 6. Then apply +6^2 = +36: 246 + 36 = 282 (matches the fifth term). 7. Finally apply -5^2 = -25: 282 - 25 = 257 (matches the last term). 8. So the pattern using successive squares is perfectly consistent when the missing term is 246. Verification / Alternative check: Write out all operations clearly: 312 - 81 = 231. 231 + 64 = 295. 295 - 49 = 246. 246 + 36 = 282. 282 - 25 = 257. The square sequence used is 9^2, 8^2, 7^2, 6^2, 5^2 with alternating minus and plus signs, which is a neat and typical exam pattern. Why Other Options Are Wrong: 210, 25, 279: Substituting any of these for the missing term breaks the square based pattern and does not yield exactly -7^2 and +6^2 operations between the surrounding terms. Common Pitfalls: Students may focus on approximate differences and miss the link to perfect squares. Another pitfall is to assume a constant difference or ratio. When differences are large and irregular, it is often beneficial to check for square or cube patterns, especially if values like 81, 64 and 25 appear. Final Answer: The number that should replace the question mark is 246.

Question 19

Choose the correct alternative that will continue the same number pattern and replace the missing term x in the series 4, 9, 25, x, 121, 169, 289, 361.

Accepted Answer

Introduction / Context: This question tests number series reasoning where each term follows a hidden mathematical rule. Learners must identify the pattern across the entire sequence, not just locally between two numbers. The focus here is on recognizing perfect squares linked with prime numbers and then using that pattern to determine the missing term x. Such questions appear very frequently in aptitude tests, banking examinations, and other competitive exams where numerical logic and observation skills are evaluated. Given Data / Assumptions: The given number series is 4, 9, 25, x, 121, 169, 289, 361. Exactly one term x is missing, and we must replace it using the same pattern that generates the other terms. All terms are positive integers and appear to be perfect squares. Only one of the given answer options will fit the underlying rule consistently. Concept / Approach: The best way to approach this series is to look at the terms as possible squares of integers, and then see if those integers follow some special pattern. If we can express each term as n^2, the base numbers may follow a sequence like natural numbers, prime numbers, odd numbers, or some other standard pattern. Once we identify the pattern in the base numbers, we can square the missing base number to get the required value of x. This approach converts the original series problem into a simpler pattern recognition task on smaller integers. Step-by-Step Solution: Step 1: Express each visible term as a square: 4 = 2^2, 9 = 3^2, 25 = 5^2, 121 = 11^2, 169 = 13^2, 289 = 17^2, 361 = 19^2. Step 2: Look at the base numbers: 2, 3, 5, ?, 11, 13, 17, 19. Step 3: Observe that these base numbers are prime numbers in ascending order: 2, 3, 5, 7, 11, 13, 17, 19. Step 4: The missing prime between 5 and 11 is 7. Therefore the missing term x must be 7^2. Step 5: Compute 7^2 = 49. So x = 49 is the only value that preserves the prime square pattern. Verification / Alternative check: To verify, rewrite the entire series using the identified rule: 2^2 = 4, 3^2 = 9, 5^2 = 25, 7^2 = 49, 11^2 = 121, 13^2 = 169, 17^2 = 289, 19^2 = 361. The base sequence 2, 3, 5, 7, 11, 13, 17, 19 is exactly the sequence of prime numbers from 2 to 19. There are no breaks or inconsistencies when we choose 49 as the missing term. Any other number in that position would destroy this elegant prime square pattern. This confirms that our interpretation of the series is mathematically consistent and robust. Why Other Options Are Wrong: Option 81 is 9^2, but 9 is not a prime, so it breaks the prime number pattern. Option 36 is 6^2, and 6 is composite and also not in the observed prime sequence. Option 64 is 8^2, and 8 is again not a prime. Option 100 is 10^2, but 10 is composite and does not fit into the strict prime sequence between 5 and 11. Therefore none of these alternatives maintain the consistent structure of squares of consecutive prime numbers. Common Pitfalls: A common mistake is to look only at the differences between consecutive terms and try to force a pattern with additions or subtractions, which in this case looks messy and non uniform. Another error is to assume the squares relate to ordinary consecutive integers instead of prime numbers. Some students also get misled by large numbers like 289 and 361 and stop checking whether they are squares of primes. Ignoring the nature of the base numbers and focusing only on raw values often leads to incorrect choices such as 36 or 64. Final Answer: The missing term that maintains the pattern of squares of consecutive prime numbers is 49, so the correct answer is 49.

Question 20

Choose the correct alternative that will continue the same pattern and replace the missing term x in the series 525, 430, 345, 270, x.

Accepted Answer

Introduction / Context: This question involves a decreasing number series where each successive term is obtained by subtracting a regular, structured amount from the previous term. Such problems evaluate a candidate understanding of arithmetic progressions in the differences of a series. Instead of working directly with the original numbers, the focus is usually on the pattern formed by the differences between consecutive terms. Once that difference pattern is recognized, finding the missing term becomes straightforward. Given Data / Assumptions: The given series is 525, 430, 345, 270, x. We must determine the single missing term x using the same pattern observed from the earlier consecutive terms. The numbers are decreasing, which suggests subtraction with a pattern in the subtracted values. Only one option will satisfy the same rule throughout the entire series. Concept / Approach: The most natural approach here is to look at consecutive differences. When we subtract each term from the previous one, we often find that the differences themselves either form an arithmetic progression, a geometric progression, or follow some simple rule such as constant changes. After establishing the pattern in these differences, we can apply the same logic to compute the missing term x. This method is widely used in number series questions, especially when the original numbers are moderately large and clearly not squares or cubes of a simple sequence. Step-by-Step Solution: Step 1: Compute the first difference: 525 - 430 = 95. Step 2: Compute the second difference: 430 - 345 = 85. Step 3: Compute the third difference: 345 - 270 = 75. Step 4: Observe that the differences form the descending sequence 95, 85, 75, which decreases by 10 each time. Step 5: Continue this pattern: the next difference should be 65, so x = 270 - 65 = 205. Verification / Alternative check: Verify the full pattern including the found value. Starting with 525, subtract 95 to get 430. From 430, subtract 85 to get 345. From 345, subtract 75 to get 270. From 270, subtract 65 to get 205. The differences now are 95, 85, 75, 65, forming a clean arithmetic progression with common difference minus 10. There is no break or irregularity in this pattern, demonstrating that the discovered value 205 is fully consistent with the structure of the given series. Why Other Options Are Wrong: Option 255 would produce a difference of only 15 from 270, which completely breaks the sequence of differences. Option 235 gives a difference of 35 and again does not follow the minus 10 pattern. Option 175 gives a difference of 95, which would repeat the first difference and disrupt the steadily decreasing nature. Option 195 yields a difference of 75, repeating an earlier difference and failing to maintain the arithmetic progression. Therefore only 205 preserves the descending difference pattern of 95, 85, 75, 65. Common Pitfalls: Students often try to relate each term to the very first term using complex formulas instead of simply examining consecutive differences. Another common mistake is to assume multiplication or division when the numbers are decreasing, even though subtraction is clearly more natural in such a context. Some learners also check only one gap instead of verifying that the entire sequence of differences follows a uniform rule, which can lead to accepting a wrong option that fits locally but not globally. Always ensure consistency for every adjacent pair in the series. Final Answer: The missing term that maintains the consistent difference pattern 95, 85, 75, 65 is 205, so the correct answer is 205.

Number Series Questions