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

Insert the missing number. 7, 26, 63, 124, 215, 342, (....)

Accepted Answer

Numbers are (23 - 1), (33 - 1), (43 - 1), (53 - 1), (63 - 1), (73 - 1) etc. So, the next number is (83 - 1) = (512 - 1) = 511.

Question 11

Find the odd man out. 41, 43, 47, 53, 61, 71, 73, 81

Accepted Answer

Each of the numbers except 81 is a prime number.

Question 12

Insert the missing number. 8, 7, 11, 12, 14, 17, 17, 22, (....)

Accepted Answer

There are two series (8, 11, 14, 17, 20) and (7, 12, 17, 22) increasing by 3 and 5 respectively.

Question 13

Find the odd man out? 396, 462, 572, 427, 671, 264.

Accepted Answer

Here the given series is 396, 462, 572, 427, 671, 264. In all the terms, the middle digit is the sum of first and third digit except 427. So the Odd number in the given series is 427.

Question 14

Find the odd man out. 2, 5, 10, 17, 26, 37, 50, 64

Accepted Answer

(1*1)+1 , (2*2)+1 , (3*3)+1 , (4*4)+1 , (5*5)+1 , (6*6)+1 , (7*7)+1 , (8*8)+1 But, 64 is out of pattern.

Question 15

Insert the missing number. 2, 4, 12, 48, 240, (....)

Accepted Answer

Go on multiplying the given numbers by 2, 3, 4, 5, 6. So, the correct next number is 1440.

Question 16

Insert the missing number. 3, 7, 6, 5, 9, 3, 12, 1, 15, (....)

Accepted Answer

There are two series, beginning respectively with 3 and 7. In one 3 is added and in another 2 is subtracted. The next number is 1 - 2 = -1.

Question 17

Find the odd man out. 6, 9, 15, 21, 24, 28, 30

Accepted Answer

Each of the numbers except 28, is a multiple of 3.

Question 18

Insert the missing number. 2, 6, 12, 20, 30, 42, 56, (....)

Accepted Answer

The pattern is 1 x 2, 2 x 3, 3 x 4, 4 x 5, 5 x 6, 6 x 7, 7 x 8. So, the next number is 8 x 9 = 72.

Question 19

Find out the wrong number in the given sequence of numbers. 1, 2, 6, 15, 31, 56, 91

Accepted Answer

1, 1 + 12 = 2, 2 + 22 = 6, 6 + 32 = 15, 15 + 42 = 31, 31 + 52 = 56, 56 + 62 = 92 Last number of given series must be 92 not 91.

Question 20

Find the odd man out. 331, 482, 551, 263, 383, 362, 284

Accepted Answer

In each number except 383, the product of first and third digits is the middle one.

Odd Man Out and Series Questions