Question 1

Geometric progression — determine the first term: In a finite GP, the sum of the first and the last terms is 66, and the product of the second and the second-last terms is 128. Find the first term.

Accepted Answer

Introduction / Context: Let the first term be a, common ratio r, and number of terms n. Denote t = r^(n−1). The given conditions relate a and t without needing n explicitly. Given Data / Assumptions: a + a t = 66 ⇒ a(1 + t) = 66. (a r) * (a r^(n−2)) = a^2 r^(n−1) = a^2 t = 128. Concept / Approach: Eliminate r and n using t. Solve the system a(1+t) = 66 and a^2 t = 128. Step-by-Step Solution: a = 66 / (1 + t)a^2 t = 128 ⇒ (66^2 t) / (1 + t)^2 = 128Solve: 4356 t = 128(1 + 2t + t^2) ⇒ 32 t^2 − 1025 t + 32 = 0t = 32 or t = 1/32 ⇒ a = 66/(1+32) = 2, or a = 66/(1 + 1/32) = 64 Verification / Alternative check: Both a = 2 and a = 64 satisfy the relations for appropriate r and n symmetric about the sequence ends. Why Other Options Are Wrong: Singletons 64 or 2 exclude the other valid branch; 32 is not a valid first term from the derived quadratic. Common Pitfalls: Mixing “sum of first and last” with total GP sum; attempting to solve for r and n unnecessarily. Final Answer: 64 or 2

Question 2

Harmonic progression — find the 3rd term using an AP condition: The 1st term of a harmonic progression (HP) is 1/17. The product of the 2nd and 4th terms equals the product of the 5th and 6th terms. Find the 3rd term of the HP.

Accepted Answer

Introduction / Context: In an HP, reciprocals of the terms form an AP. Use that correspondence to translate the given product condition into a simple relation in the AP's first term A and common difference d. Given Data / Assumptions: HP: H_k = 1 / (A + (k−1)d). H_1 = 1/17 ⇒ A = 17. H_2 * H_4 = H_5 * H_6. Concept / Approach: Rewrite the products using AP denominators and cancel numerators to get an equation in A and d. Then compute H_3 = 1/(A + 2d). Step-by-Step Solution: H_2 H_4 = 1/[(A+d)(A+3d)], H_5 H_6 = 1/[(A+4d)(A+5d)]Equality ⇒ (A+4d)(A+5d) = (A+d)(A+3d)Expand: A^2 + 9Ad + 20d^2 = A^2 + 4Ad + 3d^2 ⇒ 5Ad + 17d^2 = 0With A = 17 ⇒ d = −5H_3 = 1/(A+2d) = 1/(17 − 10) = 1/7 Verification / Alternative check: Compute H_2, H_4, H_5, H_6 explicitly with A = 17, d = −5; the equality holds. Why Other Options Are Wrong: 1/14 and 1/35 correspond to wrong d values; they do not satisfy the given product constraint. Common Pitfalls: Forgetting to invert to an AP; mishandling the negative common difference. Final Answer: 1/7

Question 3

Series completion with alternating +30 and +60 increments: Fill the missing value that fits: 165, 195, 255, 285, ..., 435

Accepted Answer

Introduction / Context: Observe alternating increments. When a series alternates between two fixed steps, apply them in turn to bridge from the start to the end. Given Data / Assumptions: Partial series: 165, 195, 255, 285, ..., 435 Differences: +30, +60, +30, ... (alternates +30 and +60) Concept / Approach: Continue the alternation: after +30 from 255 to 285, the next step is +60, followed by +30 to reach 435. Step-by-Step Solution: 165 → 195 (+30)195 → 255 (+60)255 → 285 (+30)285 → 345 (+60)345 → 375 (+30), 375 → 435 (+60) Verification / Alternative check: Both end points are reached exactly when maintaining the +30, +60 alternating pattern. Why Other Options Are Wrong: 340, 341, 401 break the alternation or misalign the final target 435. Common Pitfalls: Assuming a constant difference; the alternation is the key. Final Answer: 345

Question 4

Odd-man-out in a near-square-difference series: Identify the term that does not fit: 16, 19, 21, 30, 46, 71, 107

Accepted Answer

Introduction / Context: The target series appears to add consecutive square numbers starting from 3^2 onward after an initial irregularity. Spotting the consistent square increments reveals which entry breaks the rule. Given Data / Assumptions: Series: 16, 19, 21, 30, 46, 71, 107 Observed differences: +3, +2, +9, +16, +25, +36 Concept / Approach: From 21 onward the differences are perfect squares in order: 9(=3^2), 16(=4^2), 25(=5^2), 36(=6^2). To be consistent, the increments before 21 should have been 1, 4 rather than 3, 2. Step-by-Step Solution: If the pattern is +1, +4, +9, +16, +25, +36, then:16 + 1 = 17 (should replace 19)17 + 4 = 2121 + 9 = 30, then +16, +25, +36 yield 46, 71, 107 Verification / Alternative check: Replacing 19 with 17 fixes the entire sequence into consecutive square jumps. Why Other Options Are Wrong: 30 and 46 comply with +9 and +16; 20 is not even in the list. Common Pitfalls: Forgetting that odd-man-out questions ask for the inconsistent existing term, not the corrected term. Final Answer: 19

Question 5

Odd-man-out in multiplicative pattern with half-step multipliers: Identify the inconsistent term in the sequence: 4, 2, 3.5, 7.5, 26.25, 118.125

Accepted Answer

Introduction / Context: The intended pattern multiplies by successive half-integers: ×0.5, ×1.5, ×2.5, ×3.5, ×4.5, ... Starting from 4, this determines each following term uniquely. The given list contains one term that violates this rule. Given Data / Assumptions: Start = 4. Multipliers should be: 0.5, 1.5, 2.5, 3.5, 4.5. Concept / Approach: Apply the multipliers stepwise and compare with the provided terms to see where the mismatch occurs. Step-by-Step Solution: 4 × 0.5 = 2 ✅2 × 1.5 = 3.0 (but list shows 3.5 ❌)3.0 × 2.5 = 7.5 ✅7.5 × 3.5 = 26.25 ✅26.25 × 4.5 = 118.125 ✅ Verification / Alternative check: All later terms align perfectly once the second term is corrected to 3.0; hence 3.5 is the odd-man-out. Why Other Options Are Wrong: 26.25 and 118.125 obey ×3.5 and ×4.5; 2 matches ×0.5 from 4. Common Pitfalls: Assuming a single common ratio (geometric progression); here ratios change by +1.0 each step. Final Answer: 3.5

Question 6

Square-increment series — find the missing term: Complete the series: 20, 21, 25, 34, 50, ?, 111

Accepted Answer

Introduction / Context: This sequence adds consecutive perfect squares to generate the next term. Once identified, simply continue the square increments to fill the blank. Given Data / Assumptions: Series: 20, 21, 25, 34, 50, ?, 111 Differences: +1, +4, +9, +16, +25, +36 Concept / Approach: Recognize that the increments are 1^2, 2^2, 3^2, 4^2, 5^2, 6^2. Insert the 5^2 step to find the missing value, then confirm the final step uses 6^2 to reach 111. Step-by-Step Solution: 50 + 5^2 = 50 + 25 = 7575 + 6^2 = 75 + 36 = 111 Verification / Alternative check: Earlier steps: 20→21 (+1), 21→25 (+4), 25→34 (+9), 34→50 (+16), consistent. Why Other Options Are Wrong: 65 or 70 would break the final sum to 111 using the 6^2 increment. Common Pitfalls: Using linear differences; here differences themselves follow a square sequence. Final Answer: 75

Question 7

Series with square-step increments of 5k: Complete the series: 62, 87, 187, 412, 812, ?

Accepted Answer

Introduction / Context: The jumps between terms are perfect squares of multiples of 5. Recognizing this gives a straightforward continuation. Given Data / Assumptions: Series: 62, 87, 187, 412, 812, ? Differences: +25, +100, +225, +400 = 5^2, 10^2, 15^2, 20^2 Concept / Approach: The next increment should be 25^2 = 625. Step-by-Step Solution: 812 + 625 = 1437 Verification / Alternative check: Initial term plus cumulative sums of (5k)^2 produces the listed numbers; adding (25)^2 fits the pattern. Why Other Options Are Wrong: 1012, 1337, 1457 correspond to adding 200, 525, or 645, none of which are the required next square-of-5 multiple. Common Pitfalls: Assuming constant or linear growing differences; the key is the squared multiples. Final Answer: 1437

Question 8

What number should replace the question mark in the series 4, 3, 4, 7, 15, ? so that the pattern in the sequence is maintained?

Accepted Answer

Introduction: This is a number series problem where you are asked to identify the pattern governing the sequence and then use it to find the missing term. Such questions assess your ability to detect numerical relationships and patterns, which is a key part of logical and quantitative reasoning. Given Data / Assumptions: Series: 4, 3, 4, 7, 15, ? We must determine the next term in the sequence. Exactly one option continues the logical pattern. Concept / Approach: We look for relationships between consecutive terms. Sometimes the pattern involves alternating operations, changing multipliers, or additive sequences. In this case, observing how each term is obtained from the previous one using a combination of multiplication and addition or subtraction reveals a consistent pattern with alternating +1 and −1 and gradually increasing multipliers. Step-by-Step Solution: Step 1: Examine consecutive terms:4 → 3, 3 → 4, 4 → 7, 7 → 15.Step 2: Express each step as a multiplication followed by +1 or −1.From 4 to 3: 4 * 1 − 1 = 3.From 3 to 4: 3 * 1 + 1 = 4.From 4 to 7: 4 * 2 − 1 = 7.From 7 to 15: 7 * 2 + 1 = 15.Step 3: Notice the pattern:The multiplier repeats for two steps (1, 1, then 2, 2, then 3, 3, and so on).The sign alternates as −1, +1, −1, +1, etc.Step 4: The next step should use multiplier 3 and sign −1, following the pattern.Step 5: Apply this to the last term: next term = 15 * 3 − 1 = 45 − 1 = 44. Verification / Alternative check: Extend the pattern further to test consistency. After 44, the next term would logically be 44 * 3 + 1 = 133, matching the rule of two operations with multiplier 3 and alternating signs. This smooth continuation supports that 44 is the correct missing term. Why Other Options Are Wrong: 42, 53, and 56 do not fit the described rule of multipliers and alternating plus/minus operations. Plugging any of these into the sequence breaks the consistent pattern. 31 also fails to follow the 3 * 15 ± 1 rule. Only 44 satisfies 15 * 3 − 1 and keeps the pattern intact. Common Pitfalls: Students may look only at differences between terms (−1, +1, +3, +8) and not recognize a clear second level pattern. Others may guess based on rough growth trends instead of identifying a precise rule. Always check for combined operations (multiplication plus or minus a constant) and consider repeating structures over pairs of terms. Final Answer: The missing term that continues the pattern is 44.

Question 9

Consider the number series: 16, 33, 65, 131, 261, ( ? ). Following the same pattern, what is the next term in this sequence?

Accepted Answer

Introduction / Context: Number series questions test your ability to recognize patterns and rules in sequences. Here, we are given a series of five numbers and asked to predict the sixth term. To solve this type of problem, you must observe how each term is related to the previous ones, often through addition, subtraction, multiplication, or a combination of these operations. Given Data / Assumptions: Given series: 16, 33, 65, 131, 261, ? We assume there is a single consistent pattern for the entire sequence. We must find the missing sixth term. Concept / Approach: A common approach is to look at the differences between consecutive terms. If the difference itself follows a pattern (for example, doubling, arithmetic progression, or alternating additions and subtractions), we can extend this pattern to find the next term. Sometimes the series might follow a rule like a(n) = 2 * a(n-1) plus or minus a small number. Step-by-Step Solution: Step 1: Compute first-level differences between consecutive terms. 16 to 33: 33 - 16 = 17. 33 to 65: 65 - 33 = 32. 65 to 131: 131 - 65 = 66. 131 to 261: 261 - 131 = 130. Step 2: List the differences: 17, 32, 66, 130. Step 3: Notice that each difference is approximately double the previous one, with a small alternating adjustment. Double 17 and subtract 2: 2 * 17 - 2 = 34 - 2 = 32. Double 32 and add 2: 2 * 32 + 2 = 64 + 2 = 66. Double 66 and subtract 2: 2 * 66 - 2 = 132 - 2 = 130. Step 4: The pattern alternates: -2, +2, -2, so the next step should be +2. Double 130 and add 2: 2 * 130 + 2 = 260 + 2 = 262. Step 5: Add this new difference to the last known term: 261 + 262 = 523. Verification / Alternative Check: Check consistency: If we write all differences as per the discovered rule, they are 17, 32, 66, 130, 262. Each can be obtained from the previous one using the alternating pattern. Since the logic holds for every step, our prediction of the next term as 523 is reliable. Why Other Options Are Wrong: 521, 613 and 721 do not fit the carefully derived difference pattern. Using any of these would break the alternating rule on the differences and would not maintain the progression we observed. Common Pitfalls: A typical mistake is to look only for simple differences or ratios and give up when a perfect arithmetic or geometric progression is not found. Many exam series use second-level patterns, such as patterns within the differences themselves. Always consider both the terms and their differences before deciding on the rule. Final Answer: The next term in the series is 523.

Question 10

In the number series 7, 26, 63, 124, 215, 342, ( ... ), each term can be expressed as one less than a perfect cube. What is the missing next term?

Accepted Answer

Introduction / Context: This question is about recognizing a pattern in a number series and using that pattern to determine the next term. Many competitive exams include series where each term is related to powers or cubes of integers. Here, the numbers are close to perfect cubes, and noticing this will quickly lead you to the correct answer. Being able to see such relationships is important for number series questions. Given Data / Assumptions: The given series is 7, 26, 63, 124, 215, 342, ( ... ). We are asked to find the next term in this sequence. The differences between consecutive terms do not form a simple arithmetic sequence. We suspect a relationship to perfect cubes or some other polynomial pattern. Concept / Approach: Start by checking whether each term is close to a perfect cube. If we examine 2^3, 3^3, 4^3, and so on, we see that 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125, 6^3 = 216, and 7^3 = 343. Comparing these cubes to the series terms, we notice that each term is one less than the corresponding cube. Once this pattern is clear, we can predict that the next term will be one less than 8^3. Step-by-Step Solution: Check the first term: 7. Since 2^3 = 8, we have 8 - 1 = 7. Check the second term: 26. Since 3^3 = 27, we have 27 - 1 = 26. Check the third term: 63. Since 4^3 = 64, we have 64 - 1 = 63. Check the fourth term: 124. Since 5^3 = 125, we have 125 - 1 = 124. Check the fifth term: 215. Since 6^3 = 216, we have 216 - 1 = 215. Check the sixth term: 342. Since 7^3 = 343, we have 343 - 1 = 342. The pattern is that the nth term is n + 1 cubed minus 1, starting from n = 1. To find the next term, take 8^3 = 512. Subtract 1 to match the pattern: 512 - 1 = 511. Therefore, the missing next term in the series is 511. Verification / Alternative Check: Another way to verify the pattern is to subtract consecutive terms and see if those differences also follow a pattern. The differences are 26 - 7 = 19, 63 - 26 = 37, 124 - 63 = 61, 215 - 124 = 91, and 342 - 215 = 127. These difference values are 19, 37, 61, 91, 127. While these are not in a simple arithmetic progression, if we examine them closely they grow in a way consistent with the difference of cubes formula. However, the direct observation that each term is one less than a perfect cube is simpler and more reliable. Since the pattern holds for all given terms, extending it to 8^3 - 1 is well justified. Why Other Options Are Wrong: Option 391: This number is not close to 8^3 and does not fit the cube minus one pattern. Option 421: This also does not correspond to any simple cube minus one and breaks the sequence pattern. Option 481: This is closer to 7^3 rather than 8^3 and does not fit the progression of increasing cubes. Option 573: This is far above 8^3 and completely inconsistent with the pattern of n^3 - 1. Common Pitfalls: Many students initially focus on the differences between successive terms and may become stuck if they do not see a simple linear pattern. Others may try to fit an arithmetic or geometric progression directly to the original terms without considering relationships to powers. It is important in number series problems to check for powers, squares, or cubes, and also to consider operations like adding or subtracting a small constant from these powers. Recognizing the 2^3, 3^3, 4^3, and so on pattern is a valuable skill. Final Answer: The missing next term in the series is 511.

Question 11

In the list of numbers 41, 43, 47, 53, 61, 71, 73, 81, which number is the odd one out?

Accepted Answer

Introduction / Context: This question asks you to identify the odd man out in a set of numbers. Typically, such problems are based on number properties such as primality, divisibility, or belonging to some specific mathematical pattern. Here, most of the numbers form a natural group based on being prime numbers, and one does not fit that group. Recognizing prime numbers quickly is a key skill for this kind of problem. Given Data / Assumptions: The numbers given are 41, 43, 47, 53, 61, 71, 73, and 81. We must choose the single number that does not share the key property common to the others. All numbers are positive integers greater than 1. Concept / Approach: The natural property to check here is whether each number is prime. A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. By testing each number for divisibility by small primes such as 2, 3, 5, 7, and 11, we can determine which numbers are prime and identify if there is one that is composite. The composite number among mostly prime numbers will be the odd man out. Step-by-Step Solution: Check 41: It is not divisible by 2, 3, 5, or 7. Thus 41 is a prime number. Check 43: It is not divisible by 2, 3, 5, or 7. Thus 43 is a prime number. Check 47: It is not divisible by 2, 3, 5, or 7. Thus 47 is a prime number. Check 53: It is not divisible by 2, 3, 5, or 7. Thus 53 is a prime number. Check 61: It is not divisible by 2, 3, 5, or 7. Thus 61 is a prime number. Check 71: It is not divisible by 2, 3, 5, or 7. Thus 71 is a prime number. Check 73: It is not divisible by 2, 3, 5, or 7. Thus 73 is a prime number. Check 81: 81 = 9 * 9 = 3^4, so it is clearly composite and not prime. Since all the other numbers are prime and 81 is composite, 81 is the odd man out. Verification / Alternative Check: To verify our conclusion, we can focus on 81 and note that it is an obvious power of 3. Being equal to 3 * 3 * 3 * 3, it has divisors 1, 3, 9, 27, and 81. In contrast, for a number like 41, any factor other than 1 and 41 would have to be less than or equal to the square root of 41, which is slightly above 6, but 41 is not divisible by 2, 3, 5, or any other integer in that range. The same reasoning works for the other listed primes. Thus our identification of 81 as the only composite number in the group is correct. Why Other Options Are Wrong: Option 41: This is a prime number and shares the same property as most of the numbers in the list. Option 61: This is also a prime number and does not stand out in terms of primality. Option 71: Another prime, so it matches the majority pattern. Option 73: This is prime as well and is not different in the sense required by the question. Common Pitfalls: A common error is to overcomplicate the pattern search and look for hidden relationships between consecutive numbers, such as their differences or sums, while missing the simple property of primality. Another pitfall is to misjudge primality by not testing divisibility properly, for example assuming that a number ending with 1 is always prime. Remember that a methodical check for small prime divisors is usually enough for numbers in this range. Final Answer: The odd man out in the list is 81.

Question 12

In the number series 8, 7, 11, 12, 14, 17, 17, 22, ( ... ), two interleaved arithmetic patterns are present. What is the next term that should replace the dots?

Accepted Answer

Introduction / Context: This series question looks irregular at first glance, but a closer look reveals two arithmetic patterns interwoven together. The sequence alternates between two sub sequences, and the next term is determined by extending the pattern of the appropriate subsequence. Recognizing interleaved progressions is an important skill for solving many series questions in aptitude tests. Given Data / Assumptions: The series given is 8, 7, 11, 12, 14, 17, 17, 22, ( ... ). We need to determine the missing next term. We suspect that odd indexed terms and even indexed terms may follow different arithmetic rules. The pattern is assumed to continue beyond the terms shown. Concept / Approach: Separate the sequence into two subsequences based on the positions of the terms. One subsequence consists of the terms at odd positions, and the other consists of the terms at even positions. Each subsequence is then inspected to see if it forms an arithmetic progression. Once these patterns are identified, the next term in the complete sequence can be obtained by continuing the appropriate subsequence. Step-by-Step Solution: Label the terms with their positions: position 1: 8, 2: 7, 3: 11, 4: 12, 5: 14, 6: 17, 7: 17, 8: 22. Take the odd position terms: positions 1, 3, 5, 7 give 8, 11, 14, 17. Observe that 8, 11, 14, 17 is an arithmetic progression with common difference 3. The next odd position term (position 9) will therefore be 17 + 3 = 20. Now check the even position terms: positions 2, 4, 6, 8 give 7, 12, 17, 22. The even position terms form an arithmetic progression with common difference 5. Thus position 10 would be 22 + 5 = 27, but the question only asks for the next term, which is position 9. Therefore, the next term to fill the blank in the given series is 20. Verification / Alternative Check: To confirm, list more terms using the discovered patterns. For odd positions, the terms are 8, 11, 14, 17, 20, ... For even positions, they are 7, 12, 17, 22, 27, ... Interleaving these gives 8 (1st), 7 (2nd), 11 (3rd), 12 (4th), 14 (5th), 17 (6th), 17 (7th), 22 (8th), 20 (9th), 27 (10th), and so on. This reproduces the original part of the series exactly and shows how the sequence continues, confirming that 20 is indeed the correct next term. Why Other Options Are Wrong: Option 22: This is already used at position 8 as an even position term and does not fit the odd position pattern which increases by 3. Option 24: Would break the odd term arithmetic progression because the difference from 17 to 24 would be 7, not 3. Option 27: This is the expected value for position 10 (even position), not for position 9 which is odd. Option 19: Does not fit either the odd position arithmetic progression or the even position progression. Common Pitfalls: A typical mistake is to calculate simple differences between consecutive terms and look for a single pattern, which does not emerge clearly because the pattern is hidden in two subsequences. Another pitfall is to misidentify which positions are odd or even, leading to incorrect conclusions about the common differences. Properly labeling the positions and then separating the sequence is the safest approach. Final Answer: The missing next term in the series is 20.

Question 13

Among the numbers 396, 462, 572, 427, 671 and 264, which number is the odd one out based on a common divisibility rule?

Accepted Answer

Introduction / Context: This odd man out question revolves around a divisibility property rather than simple size or parity. Many such questions use divisibility by a particular prime or composite number as the hidden pattern. Here, examining divisibility by 11 reveals a clear difference between most of the numbers and one special case. Given Data / Assumptions: The numbers are 396, 462, 572, 427, 671, and 264. We are asked to find the single number that does not share a common property with the others. All numbers are positive integers. We suspect a divisibility rule may be the basis for the pattern. Concept / Approach: A useful starting point is to test for divisibility by a moderately sized number that might be a common factor, such as 3, 7, 9, or 11. For 11 in particular, there is a simple rule: for a number with digits d1, d2, d3, ..., the alternating sum of digits determines divisibility. If the absolute value of (sum of digits in odd positions - sum of digits in even positions) is a multiple of 11, then the number is divisible by 11. Applying this test or direct division to each number can show which numbers share a common factor of 11. Step-by-Step Solution: Check 396 for divisibility by 11. We can also directly divide: 396 / 11 = 36, so 396 is divisible by 11. Check 462: 462 / 11 = 42, so 462 is divisible by 11. Check 572: 572 / 11 = 52 exactly, so 572 is divisible by 11. Check 671: 671 / 11 = 61 exactly, so 671 is divisible by 11. Check 264: 264 / 11 = 24, so 264 is also divisible by 11. Now check 427: 427 / 11 is not an integer and leaves a remainder. Thus 427 is not divisible by 11. Therefore, all the numbers except 427 are divisible by 11, making 427 the odd man out. Verification / Alternative Check: We can use the standard divisibility rule for 11 as an alternative check. For 396, compute (3 + 6) - 9 = 0, which is a multiple of 11, so it is divisible. For 462, compute (4 + 2) - 6 = 0, again divisible. For 572, (5 + 2) - 7 = 0. For 671, (6 + 1) - 7 = 0. For 264, (2 + 4) - 6 = 0. But for 427, (4 + 7) - 2 = 9, which is not a multiple of 11, confirming that 427 is not divisible by 11. This reinforces our conclusion that 427 is the only number that does not share divisibility by 11. Why Other Options Are Wrong: Option 671: This number is divisible by 11 (671 = 11 * 61), so it matches the common property and cannot be the odd one out. Option 462: This is 11 * 42 and clearly shares the same divisibility property. Option 264: This equals 11 * 24, again fitting the pattern. Option 396: This equals 11 * 36 and is also part of the majority group. Common Pitfalls: Some students may try to look for patterns in the differences between numbers or for other numerical tricks, which can obscure the simpler common factor property. Others may not remember the divisibility rule for 11 and overlook this clue. It is a good practice to check divisibility by small primes and by 11 in such questions. Also, relying solely on mental division without verification can lead to mistaken judgments about which numbers are divisible. Final Answer: The odd man out based on divisibility by 11 is 427.

Question 14

In the sequence 2, 5, 10, 17, 26, 37, 50, 64, which number is the odd one out?

Accepted Answer

Introduction / Context: This question asks you to find the odd man out in a number series that follows a certain rule. The pattern is not a simple arithmetic progression, but instead involves increasing differences between consecutive terms. Recognizing such difference patterns is crucial in many series questions where the terms seem irregular at first glance. Given Data / Assumptions: The series given is 2, 5, 10, 17, 26, 37, 50, 64. We need to identify a single term that does not follow the pattern that the other terms follow. The numbers are positive and increase as we move along the series. Concept / Approach: A standard technique is to look at the differences between consecutive terms. If those differences themselves follow a regular pattern, we can then test each given term to see which one violates the rule. In this series, the differences form an increasing sequence that suggests a quadratic like pattern where the difference increases by a constant amount each time. Any term that disrupts this systematic increase will be the odd one out. Step-by-Step Solution: Compute the differences between consecutive terms: 5 - 2 = 3. 10 - 5 = 5. 17 - 10 = 7. 26 - 17 = 9. 37 - 26 = 11. 50 - 37 = 13. 64 - 50 = 14. We observe that the differences are 3, 5, 7, 9, 11, 13, and 14. The first six differences form an odd number sequence: 3, 5, 7, 9, 11, 13. Following this pattern, the next difference should be 15, not 14. So the term after 50 should be 50 + 15 = 65 to continue the pattern correctly. Instead, we have 64 in the given series, which breaks the pattern. Therefore, 64 is the odd man out. Verification / Alternative Check: If we reconstruct the ideal pattern based on the rule that each difference increases by 2 starting from 3, we get differences 3, 5, 7, 9, 11, 13, 15. Starting from 2, the next terms would then be: 2 + 3 = 5, 5 + 5 = 10, 10 + 7 = 17, 17 + 9 = 26, 26 + 11 = 37, 37 + 13 = 50, and 50 + 15 = 65. Thus, the logically consistent series is 2, 5, 10, 17, 26, 37, 50, 65. This confirms that 64 does not fit and that substituting 65 would restore the pattern. Why Other Options Are Wrong: Option 50: The difference from 37 to 50 is 13, fitting perfectly into the odd differences pattern. Option 37: The difference from 26 to 37 is 11, also following the odd number sequence. Option 26: The difference from 17 to 26 is 9, consistent with the pattern. Option 10: The difference from 5 to 10 is 5, again fitting the planned sequence. Common Pitfalls: One common mistake is to treat the series as random and guess without checking the differences. Another error is to miscalculate the differences and not notice the odd number pattern in them. Some students also look for relationships between nonconsecutive terms and miss the simpler idea that consecutive differences should increase by a constant amount. Always start by calculating first differences and, if needed, second differences when analyzing such series. Final Answer: The odd man out in the series is 64.

Question 15

In the number series 2, 4, 12, 48, 240, ( ... ), each term is obtained by multiplying the previous term by increasing integers. What is the next term?

Accepted Answer

Introduction / Context: This question presents a number series where each term is obtained from the previous term by multiplication with a changing factor. Recognizing the sequence of multipliers is key to finding the next term. Such series are common in aptitude tests and help assess pattern recognition and basic multiplication skills. Given Data / Assumptions: The given series is 2, 4, 12, 48, 240, ( ... ). We are asked to find the next term in the series. The numbers grow quickly, suggesting that each term is a multiple of the previous one. Concept / Approach: To determine the pattern, calculate the ratio of each term to the previous term. If these ratios form a simple sequence, such as consecutive integers, then we can extend that pattern to find the next multiplier and thus the next term. Here, the ratios turn out to be 2, 3, 4, and 5, which suggests multiplying by the next integer, 6, for the next term. Step-by-Step Solution: Compute the ratio of the second term to the first term: 4 / 2 = 2. Compute the ratio of the third term to the second term: 12 / 4 = 3. Compute the ratio of the fourth term to the third term: 48 / 12 = 4. Compute the ratio of the fifth term to the fourth term: 240 / 48 = 5. Thus the multipliers are 2, 3, 4, and 5 in order. The pattern seems to be that each term is multiplied by the next integer: 2, then 3, then 4, then 5. Following this pattern, the next multiplier should be 6. Therefore, the next term = 240 * 6 = 1440. So the required next term in the series is 1440. Verification / Alternative Check: We can reconstruct the series using the multipliers to verify. Starting from 2, multiply by 2 to get 4. Multiply 4 by 3 to get 12. Multiply 12 by 4 to get 48. Multiply 48 by 5 to get 240. The pattern clearly holds. If we now multiply 240 by 6, we get 1440, and this fits the expected progression. There are no irregularities or breaks in the multiplier pattern, so this confirms our answer. Why Other Options Are Wrong: Option 960: This equals 240 * 4 and ignores the increasing multiplier pattern. Option 1080: This is 240 * 4.5, which does not follow integer multipliers. Option 1920: This is 240 * 8, which skips the planned multiplier 6. Option 720: This is 240 * 3 and would move backwards in the pattern of increasing multipliers. Common Pitfalls: Some students focus on differences instead of ratios, which do not give a simple pattern for this kind of series because the differences grow faster than linearly. Others may miscalculate the multipliers or fail to notice that they form the sequence 2, 3, 4, 5. Always consider both differences and ratios in multiplicative series, and double check your arithmetic when working with larger numbers. Final Answer: The next term in the series is 1440.

Question 16

In the number series 3, 7, 6, 5, 9, 3, 12, 1, 15, ( ... ), what number should come next to continue the pattern?

Accepted Answer

Introduction / Context: This number series alternates between two different patterns, one for the odd positions and another for the even positions. The numbers appear in pairs, and analyzing how each pair changes reveals the rule for the sequence. Recognizing such paired or interleaved patterns is very important for solving more complex series questions in aptitude exams. Given Data / Assumptions: The series is 3, 7, 6, 5, 9, 3, 12, 1, 15, ( ... ). We must determine the next term after 15. Observing the sequence in pairs may make the pattern clearer. We assume the same rule continues beyond the given terms. Concept / Approach: Group the terms into ordered pairs: (3, 7), (6, 5), (9, 3), (12, 1), (15, ?). In each pair, the first number appears to increase steadily, while the second number decreases steadily. If we can describe these first elements and second elements with simple rules, we can predict the missing second element in the last pair and thus find the next term in the overall series. Step-by-Step Solution: Group the terms as pairs: (3, 7), (6, 5), (9, 3), (12, 1), (15, ?). Observe the first elements of the pairs: 3, 6, 9, 12, 15. These first elements form an arithmetic progression with first term 3 and common difference 3. Now observe the second elements of the pairs: 7, 5, 3, 1. These second elements form an arithmetic progression with first term 7 and common difference -2. Following the pattern, the next first element is already given as 15, fitting 3, 6, 9, 12, 15. The next second element should be 1 - 2 = -1 to continue the sequence 7, 5, 3, 1, -1. Thus, the missing term in the overall series after 15 is -1. Verification / Alternative Check: Let us reconstruct the full series from the paired description. Using first elements 3, 6, 9, 12, 15 and second elements 7, 5, 3, 1, -1, the pairs become (3, 7), (6, 5), (9, 3), (12, 1), (15, -1). Writing them out in sequence yields 3, 7, 6, 5, 9, 3, 12, 1, 15, -1. This exactly reproduces the given part of the sequence and appends the logical next term. Since both subsequences (one increasing by 3 and one decreasing by 2) are consistent, the answer -1 is fully justified. Why Other Options Are Wrong: Option 18: This would continue the first element pattern but 18 would be the first element of a new pair, not the second element following 15. Option 13: Does not fit into the arithmetic progression of second elements 7, 5, 3, 1, which decrease by 2 each time. Option 3: This value already appeared earlier as a second element and does not continue the decreasing pattern. Option 0: This would imply a difference of -1 from the previous second element 1, breaking the consistent step of -2. Common Pitfalls: Some students may only look at overall differences between consecutive terms and fail to notice the pair structure. This leads to confusion because the differences do not form a simple pattern if treated as a single sequence. Another common mistake is to assume that the pattern should always produce positive numbers and thus reject negative values, even when the arithmetic rules clearly lead to a negative term. In exam scenarios, always follow the discovered rule even if the next term turns out to be negative. Final Answer: The next term in the series is -1.

Question 17

From the list 6, 9, 15, 21, 24, 28, 30, which number is the odd one out based on divisibility by 3?

Accepted Answer

Introduction / Context: This is another odd man out question that is based on a simple divisibility property. Most of the numbers in the list share a common feature related to divisibility by 3, and one number does not have this property. The question is designed to test your ability to quickly identify multiples of 3 by using a basic divisibility rule. Given Data / Assumptions: The numbers given are 6, 9, 15, 21, 24, 28, and 30. We must pick the one number that does not share the key property common to the others. All numbers are positive integers. Concept / Approach: Use the divisibility rule for 3: a number is divisible by 3 if the sum of its digits is a multiple of 3. For example, 24 has digit sum 2 + 4 = 6, which is divisible by 3, so 24 is a multiple of 3. Apply this rule to each number in the list to see which one fails to be divisible by 3. That number will be the odd man out based on this property. Step-by-Step Solution: Check 6: 6 divided by 3 is 2, so 6 is divisible by 3. Check 9: 9 divided by 3 is 3, so 9 is divisible by 3. Check 15: 1 + 5 = 6, and 6 is divisible by 3, so 15 is divisible by 3. Check 21: 2 + 1 = 3, and 3 is divisible by 3, so 21 is divisible by 3. Check 24: 2 + 4 = 6, and 6 is divisible by 3, so 24 is divisible by 3. Check 30: 3 + 0 = 3, and 3 is divisible by 3, so 30 is divisible by 3. Check 28: 2 + 8 = 10, and 10 is not divisible by 3, so 28 is not divisible by 3. Therefore, 28 is the only number in the list that is not a multiple of 3 and is the odd man out. Verification / Alternative Check: We can also check by straightforward division. 6 / 3 = 2, 9 / 3 = 3, 15 / 3 = 5, 21 / 3 = 7, 24 / 3 = 8, and 30 / 3 = 10, all of which are integers. However, 28 divided by 3 gives 9 with a remainder of 1, clearly showing that 28 is not divisible by 3. Since every other number on the list divides evenly by 3, this confirms that 28 is the unique exception. Why Other Options Are Wrong: Option 21: This is a multiple of 3 and shares the same divisibility property as the majority of the list. Option 24: Also a multiple of 3 and cannot be the odd one out based on this property. Option 30: Clearly divisible by 3, so it is not different in the required sense. Option 15: Another classic multiple of 3, with digit sum equal to 6. Common Pitfalls: A common mistake is to look for more complicated patterns such as prime numbers or differences between terms, instead of noticing the simple and strong property of being a multiple of 3. Another pitfall is to perform division mentally and make a small arithmetic slip, especially with larger numbers like 28 and 30. Using the digit sum rule for divisibility by 3 is usually faster and more reliable in such scenarios. Final Answer: The odd man out in the list, based on divisibility by 3, is 28.

Question 18

In the number series 2, 6, 12, 20, 30, 42, 56, ( ... ), what is the next term that should follow 56?

Accepted Answer

Introduction / Context: This question provides a series of numbers that increase in a non linear but regular way. By examining the differences between consecutive terms, we can spot a clear pattern. Such problems are designed to test your ability to detect patterns in sequences and to use first differences to understand how the series grows. Given Data / Assumptions: The series is 2, 6, 12, 20, 30, 42, 56, ( ... ). We are asked to find the next term after 56. The terms are positive and the increments between them are increasing. Concept / Approach: To uncover the pattern, calculate the difference between each pair of successive terms. If these differences themselves follow a simple rule, such as forming an arithmetic progression, we can extend that rule to predict the next difference and hence the next term. In this series, the differences turn out to be consecutive even numbers, which makes extrapolation straightforward. Step-by-Step Solution: Compute the differences between consecutive terms: 6 - 2 = 4. 12 - 6 = 6. 20 - 12 = 8. 30 - 20 = 10. 42 - 30 = 12. 56 - 42 = 14. The differences are 4, 6, 8, 10, 12, 14. These differences form an arithmetic progression with first term 4 and common difference 2. Following this pattern, the next difference should be 16. Therefore, the next term after 56 is 56 + 16 = 72. So the required next term is 72. Verification / Alternative Check: We can interpret each term as the sum of consecutive even numbers starting from 2. For example, 2 = 2, 6 = 2 + 4, 12 = 2 + 4 + 6, 20 = 2 + 4 + 6 + 8, and so on. Under this interpretation, 56 represents the sum 2 + 4 + 6 + ... + 14. The next term would then be the sum 2 + 4 + 6 + ... + 16, which is 56 + 16 = 72. This alternative view is consistent with the difference pattern and confirms that the correct next term is 72. Why Other Options Are Wrong: Option 61: This would correspond to adding a difference of only 5 to 56, which does not match the even number difference pattern. Option 64: This implies a difference of 8 from 56, breaking the steady growth of the differences. Option 70: This uses a difference of 14 again instead of increasing it to 16, disrupting the pattern of consecutive even increments. Option 68: This adds a difference of 12, which has already been used earlier in the series. Common Pitfalls: Some students might initially think the series could be quadratic or some more complex function and overlook the simple pattern in the differences. Others may miscalculate one of the differences, which can hide the clear sequence of 4, 6, 8, 10, 12, 14. It is important to compute differences carefully and then see if those differences themselves form an arithmetic progression. Final Answer: The next term in the series after 56 is 72.

Question 19

In the number sequence 1, 2, 6, 15, 31, 56, 91, which single term is incorrect because it does not follow the underlying numerical pattern?

Accepted Answer

Introduction / Context: This question tests your ability to recognize patterns in a sequence of numbers and to identify which term breaks a clear logical rule. Such questions are very common in aptitude tests because they check both observation skills and understanding of number series. Given Data / Assumptions: The sequence given is 1, 2, 6, 15, 31, 56, 91. We are told that exactly one of these terms is wrong, and we must find the incorrect term based on a consistent numerical pattern in the rest of the sequence. We assume the pattern is simple and based on basic operations such as addition or multiplication. Concept / Approach: For such series, a good starting point is to calculate the difference between consecutive terms. Often, these differences themselves follow a simple pattern (such as squares, cubes, or arithmetic progression). Once a clear pattern is found, the term that does not fit that pattern is the wrong term. Step-by-Step Solution: Step 1: Write the sequence: 1, 2, 6, 15, 31, 56, 91.Step 2: Compute consecutive differences: 2 - 1 = 1, 6 - 2 = 4, 15 - 6 = 9, 31 - 15 = 16, 56 - 31 = 25, 91 - 56 = 35.Step 3: Observe the pattern in the differences: 1, 4, 9, 16, 25 are perfect squares (1^2, 2^2, 3^2, 4^2, 5^2). The next square should be 6^2 = 36.Step 4: According to this pattern, after 56, the next term should be 56 + 36 = 92, not 91.Step 5: Therefore, 91 is the only term that breaks the rule of adding consecutive perfect squares to get the next term. Verification / Alternative check: Start again from 1 and keep adding consecutive squares: 1 + 1 = 2, 2 + 4 = 6, 6 + 9 = 15, 15 + 16 = 31, 31 + 25 = 56, 56 + 36 = 92. This reconstructed series is 1, 2, 6, 15, 31, 56, 92, which matches all terms except the last one given in the question. This confirms that the pattern is consistent and 91 is indeed wrong. Why Other Options Are Wrong: 31, 56 and 15 each fit perfectly into the pattern of adding consecutive squares. Removing any of them would break the square-difference rule in more than one place. Only replacing 91 by 92 restores a clean pattern across the entire sequence. Common Pitfalls: Students sometimes try to guess the wrong term by looking only at approximate growth or by assuming a multiplication pattern without checking differences. Another common mistake is to not compute all differences carefully, which can hide a tidy square pattern. Always check the full run of differences before deciding which term is wrong. Final Answer: The only term that does not follow the pattern of adding consecutive perfect squares is 91.

Question 20

In the sequence 582, 605, 588, 611, 634, 617, 600, which single term is wrong because it does not fit the consistent alternating pattern in the series?

Accepted Answer

Introduction / Context: This question is about identifying the wrong term in a numerical series. The sequence here is based on alternating increments and decrements, and you need to detect that underlying rule. Such pattern-detection problems are standard in competitive exams and help assess numerical reasoning ability. Given Data / Assumptions: The sequence is: 582, 605, 588, 611, 634, 617, 600. We assume that the sequence is constructed using a simple and regular pattern involving additions and subtractions of fixed numbers. Exactly one term is assumed to be incorrect. Concept / Approach: When the numbers fluctuate (go up and down), it is natural to suspect an alternating pattern: for example, adding a fixed number, then subtracting another fixed number, and so on. The key is to compute the differences between consecutive terms and see whether a simple alternation appears. Step-by-Step Solution: Step 1: Write the sequence with differences: 582 → 605 (difference +23) 605 → 588 (difference -17) 588 → 611 (difference +23) 611 → 634 (difference +23) 634 → 617 (difference -17) 617 → 600 (difference -17)Step 2: Look for a pattern in the differences. We can see +23, -17, +23, +23, -17, -17.Step 3: A natural clean pattern would be +23, -17, +23, -17, +23, -17 (alternating between +23 and -17).Step 4: Apply the ideal pattern starting from 582: 582 + 23 = 605; 605 - 17 = 588; 588 + 23 = 611; 611 - 17 = 594; 594 + 23 = 617; 617 - 17 = 600.Step 5: In this corrected series the fourth term is 594, not 634. So the term 634 is the wrong value in the given sequence. Verification / Alternative check: Notice that with 634 kept as it is, the differences become asymmetric: we have two consecutive +23 steps in the middle (+23 from 588 to 611 and +23 from 611 to 634) and then two consecutive -17 steps at the end. This breaks the neat alternation we derived. Replacing 634 by 594 restores a perfectly regular +23, -17 alternation, confirming that 634 is indeed the outlier. Why Other Options Are Wrong: 600, 611 and 605 all become consistent members of the series once we enforce the alternate +23 and -17 rule. Changing any of them would force multiple other changes, while changing 634 alone fixes the entire pattern, so those terms cannot be the unique wrong member. Common Pitfalls: Candidates may try to fit a more complicated pattern or focus only on local differences without checking the full series. Another error is to accept a pattern that works for only a portion of the sequence. Always look for a simple, global rule that fits all other terms except one. Final Answer: The only term that violates the alternating +23 and -17 pattern is 634.

Odd Man Out and Series Questions