Q: Find the missing term in the sequence: 6, __, 21, 33, 48.

Introduction / Context:We identify the pattern that generates this sequence. Many aptitude sequences use increasing differences with a fixed step. We infer the missing term consistently with later terms. Given Data / Assumptions: Sequence: 6, __, 21, 33, 48. Likely pattern: differences increasing by a constant. Concept / Approach: Let the missing term be x. Differences: x−6, 21−x, 12, 15. If differences increase by 3 each time: (x−6), (21−x), 12, 15 with step +3. Step-by-Step Solution: Assume (21−x) − (x−6) = 3 ⇒ 21 − x − x + 6 = 3 ⇒ 27 − 2x = 32x = 24 ⇒ x = 12Check: Differences = 12−6 = 6, 21−12 = 9, 33−21 = 12, 48−33 = 15 (each +3) Verification / Alternative check:The +3 increase in differences is consistent across the entire sequence after inserting 12; pattern holds strictly. Why Other Options Are Wrong: 14, 18, 20: Break the constant +3 step in differences. None of these: Not applicable since 12 works perfectly. Common Pitfalls: Guessing without checking later terms. Assuming constant difference instead of increasing difference. Final Answer:12

Q: Evaluate the alternating sum 1 − 2 + 3 − 4 + 5 − 6 + … up to 100 terms.

Introduction / Context:We sum the first 100 integers with alternating signs. Grouping consecutive pairs simplifies the computation because each pair contributes a constant amount. Given Data / Assumptions: Series: 1 − 2 + 3 − 4 + … up to 100 terms (i.e., through 100). Standard integer arithmetic; no special series formula required. Concept / Approach: Group terms in pairs: (1 − 2), (3 − 4), (5 − 6), … Each pair equals −1. There are 100/2 = 50 such pairs. Step-by-Step Solution: (1 − 2) = −1(3 − 4) = −1⋯ 50 pairs in totalSum = 50 * (−1) = −50 Verification / Alternative check:Partial sums oscillate: S_2 = −1, S_4 = −2, … S_100 = −50, confirming the pairwise method. Why Other Options Are Wrong: −150, −60, −100: Do not reflect the consistent −1 per pair across 50 pairs. None of these: Not applicable; −50 is exact. Common Pitfalls: Stopping at 99 terms (odd count) by mistake. Miscounting the number of pairs. Final Answer:−50

Q: If A, G, and H denote the arithmetic mean, geometric mean, and harmonic mean (for two positive numbers), which identity holds among them?

Introduction / Context:For two positive numbers, the classical relationships among arithmetic mean (A), geometric mean (G), and harmonic mean (H) include a useful identity that links them exactly, not just via inequalities. Given Data / Assumptions: Two positive numbers x and y. Definitions: A = (x + y)/2, G = sqrt(xy), H = 2xy/(x + y). Concept / Approach: Compute A*H using the definitions, and compare to G^2. Step-by-Step Solution: A * H = [(x + y)/2] * [2xy/(x + y)] = xyG^2 = (sqrt(xy))^2 = xyTherefore, A * H = G^2 Verification / Alternative check:Pick x = 4, y = 9: A = 6.5, G = 6, H = 2*36/13 ≈ 5.538; A*H ≈ 36 = G^2; identity holds. Why Other Options Are Wrong: A × H = G and the ratio forms do not simplify to xy consistently. None of these: Not applicable; A × H = G^2 is exact for two numbers. Common Pitfalls: Confusing the AM–GM–HM inequalities with this identity. Applying the identity to more than two numbers, where it does not generally hold. Final Answer:A × H = G^2

Q: Find the missing term in the geometric-like progression: 4, 12, 36, __, 324, 972.

Introduction / Context:This number pattern multiplies each term by a constant factor to obtain the next term. Recognizing the factor allows us to insert the missing value correctly. Given Data / Assumptions: Sequence: 4, 12, 36, __, 324, 972. Terms appear to grow by a constant ratio. Concept / Approach: Check ratio: 12/4 = 3, 36/12 = 3. Assume constant ratio 3 continues. Step-by-Step Solution: Term 4 = 36 * 3 = 108Check forward: 108 * 3 = 324, 324 * 3 = 972 Verification / Alternative check:All ratios equal 3; sequence is consistent after inserting 108. Why Other Options Are Wrong: 125, 215, 312: Do not maintain a constant ratio of 3 throughout. None of these: Not applicable; 108 fits perfectly. Common Pitfalls: Trying to apply addition instead of multiplication in geometric-like sequences. Final Answer:108

Question 1

Odd Man Out – Spot the term that breaks the descending-division pattern: 2880, 480, 92, 24, 8, 4, 2

Accepted Answer

Introduction / Context:Odd-man-out tasks often rely on a clean rule with one deliberate violation. Here the numbers suggest repeated division by descending integers: ÷6, ÷5, ÷4, ÷3, ÷2, ÷1. We inspect each step to find the mismatch. Given Data / Assumptions: Displayed: 2880, 480, 92, 24, 8, 4, 2 Hypothesis: divide by 6, then 5, then 4, then 3, then 2, then 1 Concept / Approach:Check each step numerically. If the pattern holds, 2880 ÷ 6 = 480, 480 ÷ 5 = 96 (not 92), 96 ÷ 4 = 24, 24 ÷ 3 = 8, 8 ÷ 2 = 4. The only break is at the second division. Step-by-Step Solution:2880 ÷ 6 = 480 ✔480 ÷ 5 = 96 ✖ (given 92)96 ÷ 4 = 24 ✔24 ÷ 3 = 8 ✔8 ÷ 2 = 4 ✔ Verification / Alternative check:If 92 were replaced with 96, the chain becomes perfectly consistent with descending divisors 6, 5, 4, 3, 2, 1 (ending at 4, then 4 ÷ 1 = 4). The provided tail also includes 2, which may indicate an extra halving, but the unique clear error is 92. Why Other Options Are Wrong:2880, 480, 24, and 8 fit the simple division-by-integer ladder; only 92 disrupts the exact step size. Common Pitfalls:Overlooking a single incorrect intermediate and focusing on the tail end; verify each division explicitly. Final Answer:92

Question 2

Odd Man Out – Numbers of the form (2^n − 1): 3, 7, 15, 27, 63, 127, 255

Accepted Answer

Introduction / Context:This odd-man-out test leverages the well-known pattern 2^n − 1: 3, 7, 15, 31, 63, 127, 255, … Only one term in the given list does not conform to this form, and that is the answer. Given Data / Assumptions: Given sequence: 3, 7, 15, 27, 63, 127, 255 Target form: 2^n − 1 for integer n ≥ 2 Concept / Approach:Check each term against near powers of two. 3=4−1, 7=8−1, 15=16−1, 63=64−1, 127=128−1, 255=256−1. Only 27 is not one less than a power of two (the nearest is 32, giving 31). Step-by-Step Solution:3 = 2^2 − 1 ✔7 = 2^3 − 1 ✔15 = 2^4 − 1 ✔27 ≠ 2^n − 1 (since 2^5 − 1 = 31) ✖63 = 2^6 − 1 ✔; 127 = 2^7 − 1 ✔; 255 = 2^8 − 1 ✔ Verification / Alternative check:Replacing 27 with 31 would produce a perfect run of 2^n − 1 terms. Why Other Options Are Wrong:7, 15, 63, and 127 all equal 2^n − 1 for some n; they are consistent with the rule. Common Pitfalls:Confusing 27 (which equals 3^3) with a 2-power-minus-one value. Final Answer:27

Question 3

Odd Man Out – Product-plus-index pattern check: 7, 8, 18, 57, 228, 1165, 6996

Accepted Answer

Introduction / Context:Many constructed series follow a rule of the form “multiply by k, then add k” where k increases step by step. The task is to verify each transition and find the single term that breaks the otherwise consistent pattern. Given Data / Assumptions: Sequence: 7, 8, 18, 57, 228, 1165, 6996 Hypothesis: a(n+1) = a(n) * n + n for n starting at 1 (or equivalently “×n then +n”). Concept / Approach:Test the evolving multiplier/addend by index: 7*1+1=8; 8*2+2=18; 18*3+3=57; next should be 57*4+4 = 232, not 228. After that, 232*5+5=1165; 1165*6+6=6996. Only one value breaks the chain. Step-by-Step Solution:7 → 8: 7*1 + 1 = 8 ✔8 → 18: 8*2 + 2 = 18 ✔18 → 57: 18*3 + 3 = 57 ✔57 → (should be) 57*4 + 4 = 232But given term is 228 ✖Continuing: 232*5 + 5 = 1165 ✔; 1165*6 + 6 = 6996 ✔ Verification / Alternative check:Replacing 228 with 232 restores a perfect application of the rule across all steps. Why Other Options Are Wrong:8, 18, 57, 1165 are each exactly of the form “previous*index + index” as demonstrated. Common Pitfalls:Searching for a single constant multiplier or additive when the rule evolves with the index. Final Answer:228

Question 4

Find the wrong number by inspecting the stepped-decrement pattern: 125, 106, 88, 76, 65, 58, 53

Accepted Answer

Introduction / Context:Wrong-number series require reconstructing the intended step pattern and spotting the single term that distorts it. Here, the decrements are nearly regular but one term causes a mismatch relative to a clean scheme with descending primes. Given Data / Assumptions: Series: 125, 106, 88, 76, 65, 58, 53 Consider a pattern of subtracting descending primes: 19, 17, 13, 11, 7, 5 Concept / Approach:Apply the descending prime subtractions from 125: 125−19=106; next should be 106−17=89 (but given 88). Continuing with 89−13=76, 76−11=65, 65−7=58, 58−5=53 shows a perfectly consistent path if 89 replaces 88. Step-by-Step Solution:125 − 19 = 106 ✔106 − 17 = 89 (expected); given is 88 ✖89 − 13 = 76 ✔76 − 11 = 65 ✔65 − 7 = 58 ✔; 58 − 5 = 53 ✔ Verification / Alternative check:Using the prime-stepping rule, every term fits precisely once 88 is corrected to 89. The rest of the chain remains intact, pointing uniquely to 88 as the outlier. Why Other Options Are Wrong:125, 106, 76, and 65 all align with the descending prime decrements when 88 is adjusted to 89. Common Pitfalls:Trying to force a single arithmetic difference; many wrong-number series rely on motif sequences like descending primes. Final Answer:88

Question 5

Odd Man Out – Verify the “add perfect squares” motif: 1, 5, 14, 30, 50, 55, 91

Accepted Answer

Introduction / Context:Here the natural hypothesis is “add consecutive square numbers.” This is common in reasoning tests. We test each jump to see whether it equals 2^2, 3^2, 4^2, 5^2, 6^2, … The single break identifies the odd term. Given Data / Assumptions: Series: 1, 5, 14, 30, 50, 55, 91 Target differences: 4, 9, 16, 25, 36, … Concept / Approach:Compute differences: 1→5 (+4=2^2); 5→14 (+9=3^2); 14→30 (+16=4^2); next should be +25 to reach 55. Hence 30→55 is correct with +25, but 50 is not on this path, revealing it as the odd term. Step-by-Step Solution:1 + 4 = 5 ✔5 + 9 = 14 ✔14 + 16 = 30 ✔Expected next: 30 + 25 = 55 (not 50) ✖Continuing: 55 + 36 = 91 ✔ Verification / Alternative check:Replacing 50 with 55 preserves the “add squares of 2,3,4,5,6” chain perfectly. Why Other Options Are Wrong:5, 55, 91, and 14 all sit exactly where adding perfect squares dictates. Common Pitfalls:Missing that 55 appears later, correctly placed, and thinking it duplicates 50’s role; actually 50 is the anomaly. Final Answer:50

Question 6

Find the next prime in the sequence of odd primes: 7, 11, 13, 17, 19, …

Accepted Answer

Introduction / Context:This is a prime-number recognition problem. The sequence lists consecutive odd primes in ascending order. We need the next prime after 19. Given Data / Assumptions: Sequence: 7, 11, 13, 17, 19. All terms shown are prime numbers and increase. Concept / Approach: Prime numbers are integers greater than 1 with no positive divisors other than 1 and themselves. Check the next odd integers after 19: 21 (composite), 23 (prime). Step-by-Step Solution: After 19, test 21: divisible by 3 and 7 ⇒ not prime.Next 23: not divisible by 2, 3, or 5; √23 < 5, so only small primes need testing ⇒ prime. Verification / Alternative check:Known list of primes confirms 23 follows 19 in the odd prime sequence after excluding 21 as composite. Why Other Options Are Wrong: 21 and 27 are composite (21 = 3*7; 27 = 3^3). None of these: Incorrect because 23 is the correct next prime. Common Pitfalls: Assuming every odd number is prime. Stopping checks too early without testing divisibility by small primes. Final Answer:23

Question 7

Find the missing term in the sequence: 6, __, 21, 33, 48.

Accepted Answer

Introduction / Context:We identify the pattern that generates this sequence. Many aptitude sequences use increasing differences with a fixed step. We infer the missing term consistently with later terms. Given Data / Assumptions: Sequence: 6, __, 21, 33, 48. Likely pattern: differences increasing by a constant. Concept / Approach: Let the missing term be x. Differences: x−6, 21−x, 12, 15. If differences increase by 3 each time: (x−6), (21−x), 12, 15 with step +3. Step-by-Step Solution: Assume (21−x) − (x−6) = 3 ⇒ 21 − x − x + 6 = 3 ⇒ 27 − 2x = 32x = 24 ⇒ x = 12Check: Differences = 12−6 = 6, 21−12 = 9, 33−21 = 12, 48−33 = 15 (each +3) Verification / Alternative check:The +3 increase in differences is consistent across the entire sequence after inserting 12; pattern holds strictly. Why Other Options Are Wrong: 14, 18, 20: Break the constant +3 step in differences. None of these: Not applicable since 12 works perfectly. Common Pitfalls: Guessing without checking later terms. Assuming constant difference instead of increasing difference. Final Answer:12

Question 8

Evaluate the alternating sum 1 − 2 + 3 − 4 + 5 − 6 + … up to 100 terms.

Accepted Answer

Introduction / Context:We sum the first 100 integers with alternating signs. Grouping consecutive pairs simplifies the computation because each pair contributes a constant amount. Given Data / Assumptions: Series: 1 − 2 + 3 − 4 + … up to 100 terms (i.e., through 100). Standard integer arithmetic; no special series formula required. Concept / Approach: Group terms in pairs: (1 − 2), (3 − 4), (5 − 6), … Each pair equals −1. There are 100/2 = 50 such pairs. Step-by-Step Solution: (1 − 2) = −1(3 − 4) = −1⋯ 50 pairs in totalSum = 50 * (−1) = −50 Verification / Alternative check:Partial sums oscillate: S_2 = −1, S_4 = −2, … S_100 = −50, confirming the pairwise method. Why Other Options Are Wrong: −150, −60, −100: Do not reflect the consistent −1 per pair across 50 pairs. None of these: Not applicable; −50 is exact. Common Pitfalls: Stopping at 99 terms (odd count) by mistake. Miscounting the number of pairs. Final Answer:−50

Question 9

If A, G, and H denote the arithmetic mean, geometric mean, and harmonic mean (for two positive numbers), which identity holds among them?

Accepted Answer

Introduction / Context:For two positive numbers, the classical relationships among arithmetic mean (A), geometric mean (G), and harmonic mean (H) include a useful identity that links them exactly, not just via inequalities. Given Data / Assumptions: Two positive numbers x and y. Definitions: A = (x + y)/2, G = sqrt(xy), H = 2xy/(x + y). Concept / Approach: Compute A*H using the definitions, and compare to G^2. Step-by-Step Solution: A * H = [(x + y)/2] * [2xy/(x + y)] = xyG^2 = (sqrt(xy))^2 = xyTherefore, A * H = G^2 Verification / Alternative check:Pick x = 4, y = 9: A = 6.5, G = 6, H = 2*36/13 ≈ 5.538; A*H ≈ 36 = G^2; identity holds. Why Other Options Are Wrong: A × H = G and the ratio forms do not simplify to xy consistently. None of these: Not applicable; A × H = G^2 is exact for two numbers. Common Pitfalls: Confusing the AM–GM–HM inequalities with this identity. Applying the identity to more than two numbers, where it does not generally hold. Final Answer:A × H = G^2

Question 10

Find the missing term in the geometric-like progression: 4, 12, 36, __, 324, 972.

Accepted Answer

Introduction / Context:This number pattern multiplies each term by a constant factor to obtain the next term. Recognizing the factor allows us to insert the missing value correctly. Given Data / Assumptions: Sequence: 4, 12, 36, __, 324, 972. Terms appear to grow by a constant ratio. Concept / Approach: Check ratio: 12/4 = 3, 36/12 = 3. Assume constant ratio 3 continues. Step-by-Step Solution: Term 4 = 36 * 3 = 108Check forward: 108 * 3 = 324, 324 * 3 = 972 Verification / Alternative check:All ratios equal 3; sequence is consistent after inserting 108. Why Other Options Are Wrong: 125, 215, 312: Do not maintain a constant ratio of 3 throughout. None of these: Not applicable; 108 fits perfectly. Common Pitfalls: Trying to apply addition instead of multiplication in geometric-like sequences. Final Answer:108

Question 11

Complete the sequence of odd squares: __, 9, 25, 49, 81, 121.

Accepted Answer

Introduction / Context:The sequence lists squares of consecutive odd numbers. Identifying this property lets us find the term preceding 9 (which is 3^2). Given Data / Assumptions: Known terms: 9, 25, 49, 81, 121 = 3^2, 5^2, 7^2, 9^2, 11^2. Concept / Approach: Preceding odd number is 1; its square is 1. Step-by-Step Solution: Odd numbers: 1, 3, 5, 7, 9, 11Squares: 1, 9, 25, 49, 81, 121Hence the missing initial term is 1 Verification / Alternative check:Sequence strictly follows (2n−1)^2; with n = 1 gives 1, matching the missing place. Why Other Options Are Wrong: 3,4,5 are not squares of an odd integer preceding 3^2 in this pattern. None of these: Not applicable because 1 is valid. Common Pitfalls: Assuming arithmetic progression of the values instead of recognizing squared odds. Final Answer:1

Question 12

Find the next term in the sequence: 12, 15, 21, 30, 42, 57, __.

Accepted Answer

Introduction / Context:We determine the rule generating the sequence. The increments between consecutive terms are increasing by a constant amount, a common pattern in aptitude tests. Given Data / Assumptions: Sequence: 12, 15, 21, 30, 42, 57, __. Concept / Approach: Compute consecutive differences and look for a simple pattern (e.g., +3, +6, +9 …). Step-by-Step Solution: 15−12 = 321−15 = 630−21 = 942−30 = 1257−42 = 15Pattern: differences increase by +3 each time ⇒ next difference = 18Next term = 57 + 18 = 75 Verification / Alternative check:Continuing the +3 step keeps the structure consistent; 75 fits naturally as the next term. Why Other Options Are Wrong: 59, 63, 65: Do not match the +3 ladder of differences. None of these: Not applicable; 75 is correct. Common Pitfalls: Assuming constant difference or ratio instead of a second-level arithmetic pattern. Final Answer:75

Question 13

Number series: 3, 5, 9, 17, 33, 65, ? Identify the next term by analyzing the pattern of successive differences.

Accepted Answer

Introduction / Context:Many number-series questions are built on differences or ratios that themselves follow a pattern. Here the terms grow moderately fast, suggesting either multiplicative growth or additive jumps that expand regularly. Given Data / Assumptions: Series: 3, 5, 9, 17, 33, 65, ? We assume a single-rule deterministic pattern applies to all steps. Concept / Approach:Compute successive differences and then look for a clear, simple law (e.g., constant, arithmetic progression, geometric progression, powers of 2). If the first differences themselves form a recognizable series, extend that series one step to project the next main term. Step-by-Step Solution:Differences: 5−3 = 2, 9−5 = 4, 17−9 = 8, 33−17 = 16, 65−33 = 32.These differences are 2, 4, 8, 16, 32, i.e., powers of 2 doubling each time.Next difference = 64.Next term = 65 + 64 = 129. Verification / Alternative check:Project backwards: starting at 3, repeatedly add 2^n (n = 1, 2, 3, 4, 5, 6). This rebuilds the sequence through 65 and yields 129 next, confirming consistency. Why Other Options Are Wrong:128, 126, and 132 ignore the doubling-difference structure. 128 would require adding 63, which breaks the powers-of-two pattern. Common Pitfalls:Trying to fit a direct multiplicative pattern (e.g., times something plus/minus) when a simple difference pattern exists; stopping after checking only one or two gaps. Final Answer:129

Question 14

Number series with a functional step: 48, 23, ?, 4.25, 1.125 Find the missing third term by identifying the rule applied at each step.

Accepted Answer

Introduction / Context:Series sometimes evolve by repeatedly applying the same function rather than adding a constant difference. Here values drop sharply, hinting at a halving or fractional operation plus a small adjustment. Given Data / Assumptions: Known terms: 48, 23, ?, 4.25, 1.125 We search for a consistent term-to-term map f(x) that fits all given steps. Concept / Approach:Test simple transformations such as x/2, x/2 ± 1, or (x ± c)/k. Check whether one rule fits both 48 → 23 and 4.25 → 1.125. If so, apply it to 23 to recover the missing middle term. Step-by-Step Solution:Check 48 → 23 via x/2 − 1: 48/2 − 1 = 24 − 1 = 23 (fits).Check 4.25 → 1.125 via x/2 − 1: 4.25/2 − 1 = 2.125 − 1 = 1.125 (fits).Apply to 23 to get the missing term: 23/2 − 1 = 11.5 − 1 = 10.5. Verification / Alternative check:Forward: 10.5/2 − 1 = 5.25 − 1 = 4.25, consistent with the displayed continuation. Why Other Options Are Wrong:10, 2.5, and 11 do not satisfy the same function across all the checked steps; they would break the uniform mapping. Common Pitfalls:Assuming pure halving (x/2) without the small offset; testing only one transition and not validating against others. Final Answer:10.5

Question 15

Rule-based growth: 4, 25, 130, ?, 3280 Identify the missing term by detecting a consistent “multiply-and-add” pattern.

Accepted Answer

Introduction / Context:Some series advance by applying the same affine transformation (multiply by a constant, then add a constant). Recognizing this quickly turns the puzzle into a straightforward computation. Given Data / Assumptions: Series: 4, 25, 130, ?, 3280 A uniform step of the form next = a*current + b is suspected. Concept / Approach:Try small integer multipliers (e.g., 3, 4, 5) and small additive constants. Validate on the first two jumps; if it matches exactly, extrapolate to the missing middle term and confirm by hitting the final term given. Step-by-Step Solution:Test *5 + 5: 4*5 + 5 = 25 (fits); 25*5 + 5 = 130 (fits).Apply to find the missing term: 130*5 + 5 = 655.Confirm last step: 655*5 + 5 = 3280 (fits). Verification / Alternative check:Any alternative affine rule must match three transitions perfectly; *5 + 5 uniquely satisfies all. Why Other Options Are Wrong:645, 678, and 685 fail to produce 3280 under the same rule on the next step. Common Pitfalls:Switching rules mid-sequence or trying to force non-integers when a simple small-integer pattern works cleanly. Final Answer:655

Question 16

Perfect squares of consecutive odd integers: 1, 9, 25, 49, 81, ? Find the next term.

Accepted Answer

Introduction / Context:The given list looks like perfect squares: 1 = 1^2, 9 = 3^2, 25 = 5^2, 49 = 7^2, 81 = 9^2. Recognizing this establishes a crystal-clear continuation rule. Given Data / Assumptions: Sequence: 1, 9, 25, 49, 81. Indices of odd integers: 1, 3, 5, 7, 9. Concept / Approach:The next odd integer after 9 is 11, so the next perfect square is 11^2. This is the simplest interpretation consistent with all provided terms. Step-by-Step Solution:Map terms to odd squares: 1^2, 3^2, 5^2, 7^2, 9^2.Next odd integer = 11.Next term = 11^2 = 121. Verification / Alternative check:No alternative simple pattern (e.g., constant differences) fits as cleanly: differences 8, 16, 24, 32 already reflect the (2n−1)-square structure. Why Other Options Are Wrong:90, 125, 135 are not perfect squares; selecting these would break the square-of-odd rule evident in the data. Common Pitfalls:Stopping at differences and missing the well-known odd-squares identity: the difference between consecutive odd squares increases by 8 each time. Final Answer:121

Question 17

Number series: 2, 15, 41, 80, 132, ? Find the next term by studying second-level patterns (increments of increments).

Accepted Answer

Introduction / Context:When first differences are not constant but still show a simple structure, the second-level pattern (how the differences change) often reveals the rule. Given Data / Assumptions: Series: 2, 15, 41, 80, 132, ? We examine differences between consecutive terms. Concept / Approach:Compute consecutive differences and look for a linear pattern (e.g., +k, +2k, +3k, …). Then extend that pattern to project the next main term. Step-by-Step Solution:Differences: 15−2 = 13; 41−15 = 26; 80−41 = 39; 132−80 = 52.These are 13, 26, 39, 52 = 1*13, 2*13, 3*13, 4*13.Next difference = 5*13 = 65.Next term = 132 + 65 = 197. Verification / Alternative check:The sequence can be reconstructed from 2 by repeatedly adding 13n for n = 1,2,3,4,5…, which unambiguously yields 197 next. Why Other Options Are Wrong:150, 178, 180 are not 65 more than 132; adopting them would break the clear multiples-of-13 pattern. Common Pitfalls:Assuming a geometric ratio or adding a fixed number, missing the structured increment growth (second-level linearity). Final Answer:197

Question 18

Consistent divisor rule: 1220, 244, __, 9.76, 1.952 Determine the missing third term that preserves the same operation each step.

Accepted Answer

Introduction / Context:A common simple rule in series is repeated division by a fixed integer. Here, values drop by a factor that appears to be stable, creating evenly spaced exponents in base form. Given Data / Assumptions: Given: 1220 → 244 → ? → 9.76 → 1.952 We assume identical operation between successive terms. Concept / Approach:Test division by 5: 1220/5 = 244 (fits). Continue: 244/5 = 48.8 (candidate missing). Check the next links to validate the same rule. Step-by-Step Solution:After 244, missing = 244/5 = 48.8.Next: 48.8/5 = 9.76 (fits); 9.76/5 = 1.952 (fits). Verification / Alternative check:Division by any other simple integer fails at one of the steps; /5 is the only small-integer rule that hits every displayed transition exactly. Why Other Options Are Wrong:49.2 and 47 do not equal 244/5; 8 breaks the uniform /5 rule and would make the later terms inconsistent. Common Pitfalls:Jumping to /10 because of the decimals, or mixing divide-by-2 and divide-by-5 alternately without validating all steps. Final Answer:48.8

Question 19

Composite rule: 12, 15, 36, ?, 480, 2415, 14508 Find the missing fourth term given a pattern “multiply by n, then add 3n”.

Accepted Answer

Introduction / Context:Some sequences use an index-based operation that changes each step (e.g., multiply by 1, then add 3; multiply by 2, then add 6; and so on). Recognizing the dependence on the step number n is key. Given Data / Assumptions: Terms (t1 to t7): 12, 15, 36, t4, 480, 2415, 14508. We suspect an n-based rule: t_{k+1} = t_k * k + 3k for k = 1, 2, 3, … Concept / Approach:Validate the rule on early transitions; if they fit, apply it to find t4 and confirm that subsequent given terms also match under k = 4, 5, 6. Step-by-Step Solution:k = 1: 12*1 + 3 = 15 (fits).k = 2: 15*2 + 6 = 36 (fits).k = 3: 36*3 + 9 = 117 ⇒ t4 = 117.k = 4: 117*4 + 12 = 480 (fits).k = 5: 480*5 + 15 = 2415 (fits); k = 6: 2415*6 + 18 = 14508 (fits). Verification / Alternative check:The rule produces all later terms exactly, uniquely pinpointing 117 as the only valid t4. Why Other Options Are Wrong:115, 109, 121 fail to yield 480 under the k = 4 step, breaking the subsequent chain. Common Pitfalls:Assuming a constant multiplier/addend and missing the step-index dependence; not checking consistency beyond the missing term. Final Answer:117

Question 20

Arithmetic progression (AP): 3, 9, 15, 21, … Find the 15th term of the AP.

Accepted Answer

Introduction / Context:In an AP, each term increases by a fixed common difference d. The nth term formula a_n = a_1 + (n−1)d gives any target term directly. Given Data / Assumptions: a_1 = 3, d = 6 (since 9−3 = 6). We seek a_15. Concept / Approach:Apply the nth-term formula: a_n = a_1 + (n−1)d. Substitute n = 15 with the known a_1 and d. Step-by-Step Solution:a_15 = 3 + (15−1)*6 = 3 + 14*6 = 3 + 84 = 87. Verification / Alternative check:List a few terms or compute a_10 = 3 + 9*6 = 57 and add five more differences (5*6 = 30) to reach 87. Why Other Options Are Wrong:80 and 85 undercount the added differences; 90 overshoots by one extra step. Common Pitfalls:Using n*d instead of (n−1)d; miscounting from the first term. Final Answer:87

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