Question 1

Number series – constant decrement: 72, 63, 54, 45, ? Identify the next term when each step decreases by the same amount.

Accepted Answer

Introduction / Context: Here the series decreases linearly. Constant negative first differences are as common as constant positive ones and should be checked immediately. Given Data / Assumptions: 72, 63, 54, 45, ? Uniform decrement suspected. Concept / Approach: Compute a couple of differences: if they match, apply the same difference again to get the next term. Step-by-Step Solution: 63 − 72 = −954 − 63 = −945 − 54 = −9Next: 45 − 9 = 36 Verification / Alternative check: Working backward (36 + 9 = 45) supports the same constant step. Why Other Options Are Wrong: 27, 81, 90 violate the −9 constant drop. Common Pitfalls: Confusing with ratio-based decreases; here the unit drop is fixed. Final Answer: 36

Question 2

Number series – perfect cubes of even integers (one typographical outlier present): 8, 64, 216, 516, 512, 1000, 1728, ? Treat “516” as an obvious typo for 512 and continue the even-cube pattern.

Accepted Answer

Introduction / Context: The list clearly points to cubes of even integers: 2^3=8, 4^3=64, 6^3=216, 8^3=512, 10^3=1000, 12^3=1728. The “516” entry is a common transcription slip for 512 and does not alter the intended pattern. Given Data / Assumptions: Even integers in order: 2, 4, 6, 8, 10, 12, … Series terms are the cubes of those integers. Concept / Approach: After 12^3, the next even integer is 14; compute 14^3. Step-by-Step Solution: 14^3 = 14 * 14 * 14 = 196 * 14 = 2744. Verification / Alternative check: Neighbor terms 1000 (10^3) and 1728 (12^3) confirm the “even cubes” interpretation; 2744 (14^3) continues it cleanly. Why Other Options Are Wrong: 4096=16^3 skips one step; 3375=15^3 and 2197=13^3 are odd cubes not matching the even-only scheme. Common Pitfalls: Being thrown off by the single mistyped 516; always check nearby perfect powers before rejecting a pattern. Final Answer: 2744

Question 3

Number series – descending base with ascending exponent (k^(k+1)): 81, 512, 2401, 7776, 15625, 16384, ? Continue the pattern where the base decreases by 1 and the exponent increases by 1.

Accepted Answer

Introduction / Context: This elegant pattern builds perfect powers with a decreasing base and an increasing exponent, creating familiar landmarks: 3^4=81, 8^3=512, 7^4=2401, 6^5=7776, 5^6=15625, 4^7=16384. The natural continuation is 3^8 followed by 2^9, etc., depending on where you start reading pairs. Given Data / Assumptions: Observed chain after 4^7=16384 is 3^8 then 2^9. Among the choices, only 512 equals 2^9. Concept / Approach: Track (base, exponent) as (8,3), (7,4), (6,5), (5,6), (4,7), (3,8), (2,9). Select the next term compatible with that ordered pair. Step-by-Step Solution: Compute 2^9 = 512. Verification / Alternative check: The uniform (−1, +1) shift in (base, exponent) between successive terms confirms the rule. Why Other Options Are Wrong: 2187=3^7, 19683=3^9, 729=3^6 do not match the required (2,9) pair for the immediate next term. Common Pitfalls: Assuming a single fixed base; here both base and exponent evolve. Final Answer: 512

Question 4

Number series – multiply by 5, then alternate “+4, −4”: 6, 26, 134, 666, 3334, 16666, ? Apply the established ×5 and alternating addition/subtraction of 4 to find the next term.

Accepted Answer

Introduction / Context: This pattern weaves a strong multiplicative jump (×5) with a tiny alternating adjuster (+4, −4). Such alternations are common distractors that you can verify quickly by checking sign flips at each step. Given Data / Assumptions: 6 → 26 (= 6*5 − 4) 26 → 134 (= 26*5 + 4) 134 → 666 (= 134*5 − 4) 666 → 3334 (= 666*5 + 4) 3334 → 16666 (= 3334*5 − 4) Concept / Approach: Continue with ×5 then flip the sign again (now +4). Step-by-Step Solution: 16666*5 = 83330.Apply +4 (alternation after previous −4): 83330 + 4 = 83334. Verification / Alternative check: Every step obeys ×5 with alternating ±4, so the extension is unambiguous. Why Other Options Are Wrong: All other numbers deviate from ×5 ± 4 by much larger margins. Common Pitfalls: Missing the small ±4 tweak and assuming pure ×5, which would be off by 4 each time. Final Answer: 83334

Question 5

Number series – multiply by 1.5, 2, 2.5, 3, 3.5, 4, ... (factors rise by 0.5): 16, 24, 48, 120, 360, 1260, ? Continue the half-step growth in multipliers.

Accepted Answer

Introduction / Context: Here the sequence grows via changing multiplicative factors that increase by 0.5 each time: ×1.5, ×2.0, ×2.5, ×3.0, ×3.5, and so on. Such “laddered multipliers” are a frequent pattern. Given Data / Assumptions: 16 → 24 (×1.5) 24 → 48 (×2.0) 48 → 120 (×2.5) 120 → 360 (×3.0) 360 → 1260 (×3.5) Concept / Approach: The next multiplier is ×4.0. Apply it to 1260. Step-by-Step Solution: 1260 × 4.0 = 5040. Verification / Alternative check: Sequence of multipliers (1.5, 2.0, 2.5, 3.0, 3.5, 4.0) is perfectly regular. Why Other Options Are Wrong: 3780, 4725, 5355 do not equal 1260×4 and would break the 0.5 step in multipliers. Common Pitfalls: Mistaking these for factorial products; while 5040 equals 7!, the generative rule here is the multiplier ladder. Final Answer: 5040

Question 6

Number series – squares of increasing primes (next prime’s square): 529, 841, 961, 1369, 1681, 1849, ? Treat each term as p^2 for p prime and extend with the next prime available in the options.

Accepted Answer

Introduction / Context: The given numbers are perfect squares that align with successive primes: 23^2=529, 29^2=841, 31^2=961, 37^2=1369, 41^2=1681, 43^2=1849. The natural continuation would be 47^2=2209, followed by 53^2=2809, etc. Since 2209 is not among the options provided, we select the next prime square that is present. Given Data / Assumptions: Primes used so far: 23, 29, 31, 37, 41, 43. Options include 2809 (53^2) and 3481 (59^2), but not 2209 (47^2). Concept / Approach: Under the Recovery-First policy, preserve the “prime squares” rule and pick the next available prime’s square from the choices. Step-by-Step Solution: Next primes after 43 are 47, 53, 59, …Among options, 2809 = 53^2 is the earliest prime square offered. Verification / Alternative check: All listed terms are squares of primes; 2809 maintains that property and minimal forward jump given the choices. Why Other Options Are Wrong: 2601=51^2 and 3249=57^2 are not prime squares; 3481=59^2 skips further ahead than needed. Common Pitfalls: Insisting on 47^2; when the exact continuation is absent from choices, select the nearest valid continuation consistent with the rule. Final Answer: 2809

Question 7

Number series – add successive perfect squares: 20, 24, 33, 49, 74, 110, ? Differences are 4, 9, 16, 25, 36; continue with the next square.

Accepted Answer

Introduction / Context: Adding consecutive perfect squares (2^2, 3^2, 4^2, …) to generate the next term is a popular test pattern. Once spotted, extension is immediate. Given Data / Assumptions: 20→24 (+4), 24→33 (+9), 33→49 (+16), 49→74 (+25), 74→110 (+36). Concept / Approach: These increments are squares of 2, 3, 4, 5, 6. The next increment must be 7^2=49. Step-by-Step Solution: Compute next term: 110 + 49 = 159. Verification / Alternative check: Listing the differences (4, 9, 16, 25, 36, 49) confirms the simple rule. Why Other Options Are Wrong: 133, 147, 163 would correspond to adding 23, 37, or 53—none continues the perfect-square ladder. Common Pitfalls: Stopping at first differences without recognizing they themselves form a classic square sequence. Final Answer: 159

Question 8

Number series – constant decrement of 121 (11^2): 796, 675, 554, 433, ? Continue subtracting 121 each time.

Accepted Answer

Introduction / Context: A fixed subtraction pattern, especially by a notable square like 121 (=11^2), is common in reasoning sets. Confirm the fixed drop and extend once more. Given Data / Assumptions: 796→675 (−121), 675→554 (−121), 554→433 (−121). Concept / Approach: Apply the same subtraction again. Step-by-Step Solution: 433 − 121 = 312. Verification / Alternative check: Adding back 121 from 312 returns to 433, confirming consistency. Why Other Options Are Wrong: 231, 213, 512 do not equal 433 − 121 and would break the uniform decrement. Common Pitfalls: Searching for changing differences where a single constant drop fits perfectly. Final Answer: 312

Question 9

Number series – add a constant 111 each time: 231, 342, 453, 564, ? Find the next term by continuing the fixed increment.

Accepted Answer

Introduction / Context: Here the series grows by a fixed amount, +111, per step. Sequences that use “repeated 1s” as a difference are a common exam trope and quick to compute. Given Data / Assumptions: 231→342 (+111), 342→453 (+111), 453→564 (+111). Concept / Approach: Apply the same increment. Step-by-Step Solution: 564 + 111 = 675. Verification / Alternative check: The linearity holds across all steps; reversing would bring 675 back to 564 via −111. Why Other Options Are Wrong: 576, 475, 567 fail the +111 rule. Common Pitfalls: Overcomplicating with multiplicative guesses; always check for fixed differences first. Final Answer: 675

Question 10

Number series – differences increase by a fixed step of +123: 499, 622, 868, 1237, 1729, 2344, ? Use the linear growth in first differences to find the next term.

Accepted Answer

Introduction / Context: Some series grow with first differences themselves forming an arithmetic progression. Here, each new difference rises by +123, creating a predictable ladder. Given Data / Assumptions: Differences: 622−499=123, 868−622=246, 1237−868=369, 1729−1237=492, 2344−1729=615. Concept / Approach: Each step adds another 123 to the previous difference. Therefore the next difference is 615 + 123 = 738. Step-by-Step Solution: Next term = 2344 + 738 = 3082. Verification / Alternative check: Back-check: 3082 − 2344 = 738 fits the +123 ladder. Why Other Options Are Wrong: 3205, 2959, 3462 do not produce the correct next difference of 738. Common Pitfalls: Missing the pattern in differences and attempting ratios, which are irregular here. Final Answer: 3082

Question 11

Number series – “×4 then subtract 1, 2, 3, 4, 5, …”: 8, 31, 122, 485, 1936, 7739, ? Continue the rule to obtain the next term.

Accepted Answer

Introduction / Context: This pattern couples a strong multiplication (×4) with a steadily growing subtraction (−1, −2, −3, …). Such “multiply then adjust” constructions are frequently used to test multi-step pattern recognition under time pressure. Given Data / Assumptions: 8 → 31 = 8*4 − 1 31 → 122 = 31*4 − 2 122 → 485 = 122*4 − 3 485 → 1936 = 485*4 − 4 1936 → 7739 = 1936*4 − 5 Concept / Approach: Next step uses −6 after ×4. Step-by-Step Solution: 7739*4 = 30956.Subtract 6: 30956 − 6 = 30950. Verification / Alternative check: Each subtraction increment increases by 1, and the ×4 factor is constant across steps—both conditions satisfied. Why Other Options Are Wrong: 46430, 34650, 42850 correspond to different adjustments and break the −6 continuation. Common Pitfalls: Assuming pure ×4; always check for small additive/subtractive ladders layered on a big multiplier. Final Answer: 30950

Question 12

Number series – find the wrong term (add consecutive odd cubes): 274, 301, 426, 769, 1498, 2824, 5026 Exactly one term breaks the pattern “next = previous + k^3” with k = 3, 5, 7, 9, 11, 13, … Identify the incorrect entry.

Accepted Answer

Introduction / Context: “Find the wrong term” problems require detecting a clean rule that all but one term obeys. A neat and test-friendly rule here is adding consecutive odd cubes to progress through the sequence, a variant that rapidly increases numbers while staying simple to verify. Given Data / Assumptions: Proposed rule: add 3^3, 5^3, 7^3, 9^3, 11^3, 13^3, … to get successive terms. Start: 274. Concept / Approach: Check each step against the odd-cube increments; any single mismatch flags the wrong term, which we minimally correct under the Recovery-First policy. Step-by-Step Solution: 274 + 3^3 = 274 + 27 = 301 ✓301 + 5^3 = 301 + 125 = 426 ✓426 + 7^3 = 426 + 343 = 769 ✓769 + 9^3 = 769 + 729 = 1498 ✓1498 + 11^3 = 1498 + 1331 = 2829 ✗ (listed as 2824 → off by −5)Continuing would be 2829 + 13^3 = 2829 + 2197 = 5026 ✓ Verification / Alternative check: The sequence is perfectly consistent if 2824 is replaced by 2829 (i.e., +11^3). All other steps check out exactly. Why Other Options Are Wrong: 301, 426, 769 conform to +27, +125, +343 respectively; 2824 alone breaks the +1331 step. Common Pitfalls: Missing that both 5026 and earlier terms still fit the odd-cube rule if the single typo is corrected. Final Answer: 2824

Question 13

Find the wrong term in the number series: 2, 8, 32, 148, 765, 4626, 32431. Exactly one term violates a consistent pattern; identify the incorrect term.

Accepted Answer

Introduction / Context: This is a wrong-term detection problem in a number series. We must discover a single, clean rule that fits all positions except one, and then mark the violating term. Given Data / Assumptions: Series: 2, 8, 32, 148, 765, 4626, 32431. Index n starts at 1 for the first step (from 2 to 8). Look for multiplicative growth with a small additive adjustment. Concept / Approach: A compact pattern that nearly fits is: a(n+1) = a(n) * (n+1) + (n+1)^2. We verify each transition and isolate any mismatch. Step-by-Step Solution: 2 → 8: 2*2 + 2^2 = 4 + 4 = 8 ✔8 → 32: 8*3 + 3^2 = 24 + 9 = 33 ✖ (given 32)32 → 148: 32*4 + 4^2 = 128 + 16 = 144 ✖ (given 148)148 → 765: 148*5 + 25 = 740 + 25 = 765 ✔765 → 4626: 765*6 + 36 = 4590 + 36 = 4626 ✔4626 → 32431: 4626*7 + 49 = 32382 + 49 = 32431 ✔ Verification / Alternative check: If the 3rd term were 33 instead of 32, then 33*4 + 16 = 148 (exact). Thus a single earlier slip (32 vs 33) explains both mismatches and restores a perfect rule for the tail. Why Other Options Are Wrong: Terms 8, 148, 765 all become consistent once the 3rd term is corrected from 32→33; they are not the primary error. Common Pitfalls: Blaming a later term when an earlier off-by-one causes downstream wobble. Always check if fixing one term repairs later transitions. Final Answer: 32 is the lone wrong term under a(n+1) = a(n)*(n+1) + (n+1)^2.

Question 14

Find the wrong term in the number series: 108, 54, 36, 18, 9, 6, 4. Exactly one term violates an alternating division pattern; identify the incorrect term.

Accepted Answer

Introduction / Context: We seek a neat, alternating multiplier/divisor pattern and then detect the unique violation in the sequence. Given Data / Assumptions: Series: 108, 54, 36, 18, 9, 6, 4. Try alternating operations such as ÷2 and ×2/3 (equivalently, multiply by 0.5 and 0.6667). Concept / Approach: Compute consecutive ratios to test the alternating rule: 108→54 (×0.5), 54→36 (×2/3), 36→18 (×0.5), next should be 18×(2/3)=12, and so on. Step-by-Step Solution: 108 → 54: ÷2 ✔54 → 36: ×(2/3) ✔36 → 18: ÷2 ✔18 → 9: (given) should be ×(2/3) → 12 ✖Continuing correctly: 12 → 8 (÷2) and 8 → 16/3 (×2/3), but given tail 9, 6, 4 approximates a later adjustment. Verification / Alternative check: Directly applying the discovered alternation, only the term at the 5th position should be 12; therefore 9 is the inconsistent value. Why Other Options Are Wrong: 54, 36, 18 match the alternation before the error; the final terms often look close but are consequences of carrying the wrong intermediate value. Common Pitfalls: Mistaking a propagated downstream drift for the original error; always fix the first mismatch. Final Answer: 9 is the wrong term (should be 12 under the alternating ÷2, ×2/3 rule).

Question 15

Find the wrong term in the number series: 0, 1, 9, 36, 99, 225, 441. The intended rule is “square of triangular numbers”: Tn^2 where Tn = 0,1,3,6,10,15,21… Identify the incorrect term.

Accepted Answer

Introduction / Context: Many classic series use triangular numbers Tn = n*(n+1)/2 and then apply a transformation such as squaring. Here, terms visibly align with Tn^2—except one. Given Data / Assumptions: Series: 0, 1, 9, 36, 99, 225, 441. Triangular numbers: 0, 1, 3, 6, 10, 15, 21… Squares: 0^2=0, 1^2=1, 3^2=9, 6^2=36, 10^2=100, 15^2=225, 21^2=441. Concept / Approach: Map each series term to Tn^2. All match perfectly except the 5th term. Step-by-Step Solution: 0 → 0^2 = 0 ✔1 → 1^2 = 1 ✔9 → 3^2 = 9 ✔36 → 6^2 = 36 ✔99 → should be 10^2 = 100 ✖225 → 15^2 = 225 ✔441 → 21^2 = 441 ✔ Verification / Alternative check: Once 99 is corrected to 100, the entire sequence becomes a clean Tn^2 progression without exception. Why Other Options Are Wrong: 9, 36, and 225 correctly correspond to 3^2, 6^2, and 15^2. Common Pitfalls: Overlooking the triangular-index pattern and testing random square bases; always check Tn = 0,1,3,6,10… Final Answer: 99 is the wrong term (should be 100).

Question 16

Find the wrong term in the number series: 2, 3, 6, 15, 45, 156.5, 630. The intended pattern multiplies by 1.5, 2, 2.5, 3, 3.5, 4 … Identify the incorrect term.

Accepted Answer

Introduction / Context: This wrong-term hunt features steadily increasing multipliers by 0.5 each step. We check every transition to isolate the single incorrect value. Given Data / Assumptions: Series: 2, 3, 6, 15, 45, 156.5, 630. Hypothesized rule: multiply by 1.5, then 2.0, then 2.5, then 3.0, then 3.5, then 4.0… Concept / Approach: Verify each step against the progressive-multiplier rule and see which term breaks it. Step-by-Step Solution: 2 × 1.5 = 3 ✔3 × 2 = 6 ✔6 × 2.5 = 15 ✔15 × 3 = 45 ✔45 × 3.5 = 157.5 → but series shows 156.5 ✖If corrected to 157.5, then 157.5 × 4 = 630 ✔ Verification / Alternative check: All early steps are exact; the only inconsistency is the penultimate multiplier result (156.5 vs 157.5). Fixing this restores a perfect ×1.5, ×2, ×2.5, ×3, ×3.5, ×4 pattern. Why Other Options Are Wrong: 3, 6, 15, and 45 all correctly satisfy their respective multipliers. Common Pitfalls: Assuming a different additive rule; the multiplicative ladder is standard in such progressions. Final Answer: None of the listed earlier terms are wrong; the incorrect value is 156.5, which should be 157.5.

Question 17

Find the wrong term in the number series: 5, 8, 16, 26, 50, 98, 194. A consistent rule from the beginning should be “next = 2 × previous − 2”. Identify the incorrect entry.

Accepted Answer

Introduction / Context: We test a simple affine recurrence that fits almost all steps: multiply by 2 and subtract 2. Given Data / Assumptions: Series: 5, 8, 16, 26, 50, 98, 194. Candidate rule: a(n+1) = 2*a(n) − 2. Concept / Approach: Check each transition under the rule and locate the earliest violation. Step-by-Step Solution: 5 → 8: 2*5 − 2 = 8 ✔8 → 16: 2*8 − 2 = 14 ✖ (given 16)If we correct to 14, then 14 → 26: 2*14 − 2 = 26 ✔26 → 50: 2*26 − 2 = 50 ✔50 → 98: 2*50 − 2 = 98 ✔98 → 194: 2*98 − 2 = 194 ✔ Verification / Alternative check: Only the second step fails; fixing 16 to 14 repairs the entire tail perfectly. Why Other Options Are Wrong: 26, 50, and 98 all satisfy the rule when the earlier misprint is corrected. Common Pitfalls: Mistaking downstream consistency for independent correctness; the first inconsistency typically identifies the wrong term. Final Answer: 16 is the wrong term (should be 14).

Question 18

Complete the series and evaluate A ÷ 36: 36, 216, 64.8, 388.8, A, 699.84, 209.952. Rule alternates ×6 and ×0.3 (i.e., ×3/10). Find A and then compute A ÷ 36.

Accepted Answer

Introduction / Context: This is a completion-and-evaluation series problem with a clear alternating multiplier pattern. Given Data / Assumptions: Series: 36, 216, 64.8, 388.8, A, 699.84, 209.952. Rule: ×6, then ×0.3, repeating. Concept / Approach: Apply the two-step alternation across the entire sequence, fill A, and finally compute the requested ratio. Step-by-Step Solution: 36 × 6 = 216 ✔216 × 0.3 = 64.8 ✔64.8 × 6 = 388.8 ✔388.8 × 0.3 = A = 116.64 ✔116.64 × 6 = 699.84 ✔699.84 × 0.3 = 209.952 ✔Finally, A ÷ 36 = 116.64 ÷ 36 = 3.24. Verification / Alternative check: Every adjacent ratio alternates between 6 and 0.3, confirming the correctness of A and the final division. Why Other Options Are Wrong: 0.324 and 6.139 are scale/rounding distractors; the exact arithmetic yields 3.24. Common Pitfalls: Confusing ×0.3 with ÷3; note ×0.3 equals ÷(10/3), not ÷3. Final Answer: 3.24.

Question 19

The sequence is: 4, 7, 12, 19, 28, A, 52. Differences increase by successive odd numbers (+3, +5, +7, +9, +11, +13). Compute A^2 − 4^2.

Accepted Answer

Introduction / Context: We are given an increasing sequence and an expression to evaluate after finding the missing term. The intended operation sign is exponentiation: A^2 minus 4^2. Given Data / Assumptions: Sequence: 4, 7, 12, 19, 28, A, 52. Successive differences: +3, +5, +7, +9, +11, +13 (odd numbers in order). Interpret “A 2 – 4 = ?” as A^2 − 4^2 under standard formatting recovery. Concept / Approach: Compute A using the difference pattern, then evaluate the requested expression with perfect squares. Step-by-Step Solution: 4 → 7: +3; 7 → 12: +5; 12 → 19: +7; 19 → 28: +9.Next two differences should be +11 and +13.Therefore A = 28 + 11 = 39, and 39 + 13 = 52 (fits).Compute A^2 − 4^2 = 39^2 − 16 = 1521 − 16 = 1505. Verification / Alternative check: Using odd-number differences produces exact matches at each step, confirming A = 39. Why Other Options Are Wrong: They result from misreading the exponent or using A*2 − 4; the recovered exponent form is consistent with the options. Common Pitfalls: Missing the exponent indicator and interpreting as multiplication; always check difference patterns and option magnitudes. Final Answer: 1505.

Question 20

Fill the missing letter in the letter-triples: FOX, GP?, HQZ. Observe the position-wise sequences to determine the middle missing letter.

Accepted Answer

Introduction / Context: In position-wise letter series, each column (1st, 2nd, 3rd letters) typically follows its own alphabetic progression. Given Data / Assumptions: Triples: F O X, G P ?, H Q Z. Alphabet index: A=1 … Z=26. Concept / Approach: Track each position independently to see consistent +1 or +2 progressions. Step-by-Step Solution: 1st letters: F(6) → G(7) → H(8): +1 each step.2nd letters: O(15) → P(16) → Q(17): +1 each step.3rd letters: X(24) → ? → Z(26): +1 each step implies Y(25). Verification / Alternative check: Rewriting as three parallel sequences (FGH), (OPQ), (XYZ) makes the pattern obvious. Why Other Options Are Wrong: Z would overshoot; T and W do not fit the +1 trajectory. Common Pitfalls: Looking for constant shifts across whole triples rather than column-wise shifts. Final Answer: Y.

Number Series Questions