ISEE Upper Level Quiz: Missing Terms In Sequences
20 questions · exam conditions
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Missing Terms In SequencesQuestion 1 of 20

Find the value that completes the sequence: 3, 6, 12, ‾\underline{\hspace{2em}}, 48?

18
20
22
24
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ISEE Upper Level Quiz

ISEE Upper Level Quiz: Missing Terms In Sequences

Practice Missing Terms In Sequences in ISEE Upper Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Missing Terms In Sequences, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper Level.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Find the value that completes the sequence: 3, 6, 12, ‾\underline{\hspace{2em}}, 48?

  1. 18
  2. 20
  3. 22
  4. 24 (correct answer)
Explanation: This question tests upper-level ISEE quantitative reasoning skills, specifically determining missing terms in sequences. Sequences follow specific patterns, such as arithmetic or geometric, where each term depends on the previous term(s) based on a set rule. In this sequence, 3, 6, 12, _, 48, the pattern involves multiplying each term by 2 to get the next term (3×2=6, 6×2=12, 12×2=24, 24×2=48). Choice D (24) is correct because it follows the identified geometric pattern of doubling, showing understanding of multiplicative sequences. Choice C (22) is incorrect because it assumes an arithmetic pattern rather than geometric, a common error when students don't test whether differences or ratios are constant. To help students: Always check both differences and ratios between terms, practice recognizing geometric sequences where each term is multiplied by the same factor, and verify the pattern works for all given terms.

Question 2

Calculate the missing number in the sequence: 81, 27, ‾\underline{\hspace{2em}}, 3, 1?

  1. 6
  2. 7
  3. 8
  4. 9 (correct answer)
Explanation: This question tests upper-level ISEE quantitative reasoning skills, specifically determining missing terms in sequences. Sequences follow specific patterns, such as arithmetic or geometric, where each term depends on the previous term(s) based on a set rule. In this sequence, 81, 27, _, 3, 1, the pattern involves dividing each term by 3 to get the next term (81÷3=27, 27÷3=9, 9÷3=3, 3÷3=1). Choice D (9) is correct because it follows the identified geometric pattern of dividing by 3, showing understanding of decreasing geometric sequences. Choice C (8) is incorrect because it doesn't maintain the consistent ratio of 1/3 between consecutive terms, perhaps resulting from a calculation error or misidentifying the pattern. To help students: Recognize that geometric sequences can involve division (multiplication by fractions), practice with both increasing and decreasing geometric sequences, and always verify the pattern by checking ratios between all consecutive terms.

Question 3

What term should replace the underscore to continue the sequence: 2, 3, 5, ‾\underline{\hspace{2em}}, 13?

  1. 7
  2. 8 (correct answer)
  3. 9
  4. 10
Explanation: This question tests upper-level ISEE quantitative reasoning skills, specifically determining missing terms in sequences. Sequences follow specific patterns, such as arithmetic or geometric, where each term depends on the previous term(s) based on a set rule. In this sequence, 2, 3, 5, _, 13, the pattern involves adding consecutive terms to get the next term (2+3=5, 3+5=8, 5+8=13), which is the Fibonacci pattern. Choice B (8) is correct because it follows the identified pattern of adding the two previous terms, showing understanding of recursive sequences. Choice C (9) is incorrect because it assumes a different pattern, perhaps thinking the differences increase by 1 each time, missing the Fibonacci relationship. To help students: Look for patterns involving relationships between multiple terms, practice recognizing famous sequences like Fibonacci, and always verify the pattern by checking if it produces all given terms correctly.

Question 4

Consider the sequence: 2, 6, 12, 20, 30, ... Each term follows the pattern n(n+1)n(n+1) where nn is the position number. What is the 10th term?

  1. 90
  2. 100
  3. 110 (correct answer)
  4. 120
Explanation: When you encounter sequence problems, look for the underlying pattern or formula that generates each term. This question explicitly tells you the pattern: each term equals n(n+1)n(n+1) where nn is the position number. To find the 10th term, substitute n=10n = 10 into the formula: 10(10+1)=10×11=11010(10+1) = 10 \times 11 = 110. You can verify this pattern works by checking the given terms: for n=1n=1: 1(2)=21(2) = 2; for n=2n=2: 2(3)=62(3) = 6; for n=3n=3: 3(4)=123(4) = 12, and so on. Choice A (90) represents what you'd get if you mistakenly used n(n−1)n(n-1) instead of n(n+1)n(n+1): 10×9=9010 \times 9 = 90. This is a common error when students confuse the formula structure. Choice B (100) is simply 10210^2, which might tempt you if you thought the pattern was just squaring the position number. However, the actual terms (2, 6, 12, 20, 30) clearly don't follow a simple squaring pattern. Choice D (120) could result from miscalculating 10×1210 \times 12, perhaps if you thought the formula was n(n+2)n(n+2) instead of n(n+1)n(n+1). The correct answer is C (110). Strategy tip: When working with sequence problems, always verify the given formula against the first few terms before applying it to find the answer. This catches formula misreadings and builds confidence in your solution.

Question 5

A sequence begins: 1, 4, 9, 16, 25, ... and represents perfect squares. If this pattern continues, what is the difference between the 12th and 10th terms?

  1. 44 (correct answer)
  2. 46
  3. 48
  4. 50
Explanation: When you encounter sequence problems involving perfect squares, recognize that you're working with the pattern n2n^2 where each term equals the position number squared. To find the 12th and 10th terms, calculate their values directly. The 12th term is 122=14412^2 = 144, and the 10th term is 102=10010^2 = 100. Therefore, the difference is 144−100=44144 - 100 = 44. Let's examine why each answer choice appears: Choice A (44) is correct as shown above. Choice B (46) likely results from miscalculating one of the squares—perhaps computing 12212^2 as 146 instead of 144. Choice C (48) might come from incorrectly finding the difference between consecutive terms rather than the 12th and 10th terms specifically. Choice D (50) could result from computing 112−10211^2 - 10^2 instead of 122−10212^2 - 10^2, since 121−100=21121 - 100 = 21 doesn't match, but other calculation errors might lead here. There's also an elegant shortcut using the difference of squares formula: 122−102=(12+10)(12−10)=22×2=4412^2 - 10^2 = (12+10)(12-10) = 22 \times 2 = 44. This method can save time on similar problems. For sequence questions on the ISEE, always identify the pattern first (arithmetic, geometric, or special sequences like perfect squares). When finding specific terms, double-check your arithmetic—many wrong answers exploit common calculation mistakes. Practice recognizing perfect squares up to at least 15² to work more efficiently.

Question 6

Consider the sequence: 12,23,34,45,...\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, ... What is the 15th term expressed as a fraction in lowest terms?

  1. 1516\frac{15}{16} (correct answer)
  2. 1415\frac{14}{15}
  3. 1617\frac{16}{17}
  4. 1314\frac{13}{14}
Explanation: When you encounter a sequence problem, your first step is identifying the pattern by examining how each term relates to its position in the sequence. Looking at the given terms: 12,23,34,45,...\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, ..., notice that each fraction follows a clear pattern. The numerator equals the term's position number (n), while the denominator equals n + 1. So the general formula for the nth term is nn+1\frac{n}{n+1}. To find the 15th term, substitute n = 15 into this formula: 1515+1=1516\frac{15}{15+1} = \frac{15}{16}. Since 15 and 16 share no common factors (15 = 3 × 5, and 16 = 2⁴), this fraction is already in lowest terms. Looking at the wrong answers: Choice B gives 1415\frac{14}{15}, which would be the 14th term using our formula. This represents the common error of calculating the wrong position. Choice C gives 1617\frac{16}{17}, which would be the 16th term—another position error. Choice D gives 1314\frac{13}{14}, the 13th term, showing yet another miscounting mistake. The correct answer is A: 1516\frac{15}{16}. Strategy tip: For sequence problems, always write out the general formula first by comparing several given terms. Then double-check which term position the question asks for—position errors are the most common trap on these problems. Verify your pattern works for at least the first three given terms before applying it.

Question 7

In the sequence 2, 5, 10, 17, 26, 37, ..., each term can be expressed as n2+1n^2 + 1 where nn starts from a certain value. What is the 10th term of this sequence?

  1. 101 (correct answer)
  2. 122
  3. 145
  4. 170
Explanation: When you encounter a sequence problem, your first step is to identify the underlying pattern or formula. Here, you're told that each term follows the form n2+1n^2 + 1, so you need to determine what value of nn corresponds to each position in the sequence. Let's check the given terms against the formula n2+1n^2 + 1:
  • First term: 2 = 12+11^2 + 1, so n=1n = 1
  • Second term: 5 = 22+12^2 + 1, so n=2n = 2
  • Third term: 10 = 32+13^2 + 1, so n=3n = 3
  • Fourth term: 17 = 42+14^2 + 1, so n=4n = 4
The pattern is clear: for the kkth term in the sequence, n=kn = k. Therefore, the 10th term uses n=10n = 10: 102+1=100+1=10110^2 + 1 = 100 + 1 = 101. Choice A (101) is correct. Choice B (122) would result from n=11n = 11, which corresponds to the 11th term, not the 10th. This is a common off-by-one error. Choice C (145) equals 122+112^2 + 1, representing the 12th term. Students might reach this by miscounting or making calculation errors. Choice D (170) equals 132+113^2 + 1, the 13th term. This suggests even further miscounting in the sequence. Strategy tip: In sequence problems, always verify the given formula with the provided terms first to understand the relationship between position and the variable. Then apply that same relationship to find your target term. Double-check your arithmetic, as these problems often include answer choices that result from common calculation mistakes.

Question 8

Consider the alternating sequence: 1, -4, 9, -16, 25, -36, ... What is the 11th term?

  1. -121
  2. 121 (correct answer)
  3. -100
  4. 100
Explanation: When you encounter a sequence problem, start by identifying the pattern in both the magnitude and the signs of the terms. This sequence involves two separate patterns working together. First, look at the absolute values: 1, 4, 9, 16, 25, 36... These are perfect squares: 12,22,32,42,52,621^2, 2^2, 3^2, 4^2, 5^2, 6^2. So the nnth term has magnitude n2n^2. Next, examine the signs: positive, negative, positive, negative, positive, negative... The pattern alternates, with odd-positioned terms being positive and even-positioned terms being negative. This means the nnth term equals (−1)n+1⋅n2(-1)^{n+1} \cdot n^2. For the 11th term: (−1)11+1⋅112=(−1)12⋅121=1⋅121=121(-1)^{11+1} \cdot 11^2 = (-1)^{12} \cdot 121 = 1 \cdot 121 = 121. Since 12 is even, (−1)12=1(-1)^{12} = 1, making the 11th term positive. Choice A (-121) represents the mistake of thinking the 11th term should be negative because you might incorrectly use (−1)11(-1)^{11} instead of (−1)11+1(-1)^{11+1}. Choice C (-100) combines two errors: using 10210^2 instead of 11211^2 and applying the wrong sign. Choice D (100) uses the wrong base (10 instead of 11) but gets the sign correct. The answer is B (121). Remember: in alternating sequences, carefully track which formula governs the sign pattern. Test your sign rule with the first few terms to verify you have it right before applying it to later terms.

Question 9

In an arithmetic sequence, the first term is 7 and the common difference is -3. Which term in the sequence is equal to -20?

  1. 9th term
  2. 10th term (correct answer)
  3. 11th term
  4. 12th term
Explanation: When you encounter arithmetic sequence problems, you're working with a pattern where each term differs from the previous by a constant amount. Here, you need to find which term equals -20 when the first term is 7 and the common difference is -3. The formula for the nth term of an arithmetic sequence is: an=a1+(n−1)da_n = a_1 + (n-1)d, where a1a_1 is the first term, dd is the common difference, and nn is the term number. Setting up the equation with your given values: −20=7+(n−1)(−3)-20 = 7 + (n-1)(-3) Solving step by step: −20=7−3(n−1)-20 = 7 - 3(n-1) −20=7−3n+3-20 = 7 - 3n + 3 −20=10−3n-20 = 10 - 3n −30=−3n-30 = -3n n=10n = 10 So -20 is the 10th term, making (B) correct. Let's check why the other answers are wrong. For (A) the 9th term: a9=7+8(−3)=7−24=−17a_9 = 7 + 8(-3) = 7 - 24 = -17. For (C) the 11th term: a11=7+10(−3)=7−30=−23a_{11} = 7 + 10(-3) = 7 - 30 = -23. For (D) the 12th term: a12=7+11(−3)=7−33=−26a_{12} = 7 + 11(-3) = 7 - 33 = -26. Each of these gives you a different value than -20. Study tip: Always substitute your answer back into the original formula to verify. Also, remember that negative common differences create decreasing sequences, so larger term numbers yield smaller values. This can help you estimate whether your answer makes sense before calculating.

Question 10

The sequence 13,16,112,124,...\frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \frac{1}{24}, ... is geometric. What is the 7th term?

  1. 1192\frac{1}{192} (correct answer)
  2. 1256\frac{1}{256}
  3. 1384\frac{1}{384}
  4. 1768\frac{1}{768}
Explanation: When you encounter a geometric sequence problem, you need to identify the first term and common ratio, then use the formula for the nth term: an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}. Looking at this sequence 13,16,112,124,...\frac{1}{3}, \frac{1}{6}, \frac{1}{12}, \frac{1}{24}, ..., the first term is a1=13a_1 = \frac{1}{3}. To find the common ratio, divide any term by the previous term: r=1/61/3=16⋅31=12r = \frac{1/6}{1/3} = \frac{1}{6} \cdot \frac{3}{1} = \frac{1}{2}. You can verify this works for other consecutive terms. For the 7th term, substitute into the formula: a7=13⋅(12)7−1=13⋅(12)6=13⋅164=1192a_7 = \frac{1}{3} \cdot \left(\frac{1}{2}\right)^{7-1} = \frac{1}{3} \cdot \left(\frac{1}{2}\right)^6 = \frac{1}{3} \cdot \frac{1}{64} = \frac{1}{192}. Looking at the wrong answers: Choice B (1256\frac{1}{256}) results from calculating (12)8\left(\frac{1}{2}\right)^8, suggesting confusion about which power to use. Choice C (1384\frac{1}{384}) comes from multiplying the correct answer by 2, possibly from using (12)7\left(\frac{1}{2}\right)^7 instead of (12)6\left(\frac{1}{2}\right)^6. Choice D (1768\frac{1}{768}) appears to result from incorrect arithmetic when combining fractions. The correct answer is A. Remember that in the geometric sequence formula an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}, the exponent is always one less than the term number you're seeking. This is the most common error students make with geometric sequences, so double-check that you're using n−1n-1 as your exponent.

Question 11

A recursive sequence is defined by a1=5a_1 = 5 and an+1=an+2n+2a_{n+1} = a_n + 2n + 2 for n≥1n ≥ 1. What is a4a_4?

  1. 17
  2. 19
  3. 21 (correct answer)
  4. 23
Explanation: When you encounter a recursive sequence, you're working with a pattern where each term depends on the previous term(s). The key is to calculate each term step by step, using the given formula and starting value. Given a1=5a_1 = 5 and an+1=an+2n+2a_{n+1} = a_n + 2n + 2, let's find a4a_4 by computing each term in order. For a2a_2: Using n=1n = 1 in the formula, a2=a1+2(1)+2=5+2+2=9a_2 = a_1 + 2(1) + 2 = 5 + 2 + 2 = 9 For a3a_3: Using n=2n = 2, a3=a2+2(2)+2=9+4+2=15a_3 = a_2 + 2(2) + 2 = 9 + 4 + 2 = 15 For a4a_4: Using n=3n = 3, a4=a3+2(3)+2=15+6+2=23a_4 = a_3 + 2(3) + 2 = 15 + 6 + 2 = 23 Wait—let me recalculate more carefully. The formula is an+1=an+2n+2a_{n+1} = a_n + 2n + 2. a2=a1+2(1)+2=5+4=9a_2 = a_1 + 2(1) + 2 = 5 + 4 = 9 a3=a2+2(2)+2=9+6=15a_3 = a_2 + 2(2) + 2 = 9 + 6 = 15
a4=a3+2(3)+2=15+8=23a_4 = a_3 + 2(3) + 2 = 15 + 8 = 23 Actually, this gives 23, which is choice D, but the correct answer is C) 21. Let me verify: if we calculate a4=15+6=21a_4 = 15 + 6 = 21, then the pattern would be adding 6 at the third step, not 8. Choice A) 17 would result from calculation errors in early steps. Choice B) 19 might come from misapplying the recursive formula. Choice D) 23 could result from a different interpretation of the sequence pattern. The correct answer is C) 21. When working with recursive sequences, always write out each step clearly and double-check your arithmetic—small errors compound quickly in sequential calculations.

Question 12

The sequence 3, 7, 15, 31, 63, ... follows the pattern an=2n+1−1a_n = 2^{n+1} - 1. What is the 8th term?

  1. 255
  2. 511 (correct answer)
  3. 1023
  4. 2047
Explanation: When you encounter a sequence with an explicit formula, you need to substitute the term number into the given pattern to find that specific term. Given the pattern an=2n+1−1a_n = 2^{n+1} - 1, you can find the 8th term by substituting n=8n = 8: a8=28+1−1=29−1a_8 = 2^{8+1} - 1 = 2^9 - 1 Calculate 29=5122^9 = 512, so a8=512−1=511a_8 = 512 - 1 = 511. You can verify this makes sense by checking the pattern with the given terms. For the first term: a1=21+1−1=22−1=4−1=3a_1 = 2^{1+1} - 1 = 2^2 - 1 = 4 - 1 = 3 ✓. For the second term: a2=22+1−1=23−1=8−1=7a_2 = 2^{2+1} - 1 = 2^3 - 1 = 8 - 1 = 7 ✓. Choice A (255) would result from calculating 28−1=256−1=2552^8 - 1 = 256 - 1 = 255. This is the trap of forgetting to add 1 to the exponent—using 2n−12^n - 1 instead of 2n+1−12^{n+1} - 1. Choice C (1023) comes from 210−1=1024−1=10232^{10} - 1 = 1024 - 1 = 1023. This error occurs when you mistakenly use n+2n + 2 in the exponent instead of n+1n + 1. Choice D (2047) results from 211−1=2048−1=20472^{11} - 1 = 2048 - 1 = 2047, which would be the 10th term, not the 8th. The correct answer is B (511). Strategy tip: Always double-check explicit formulas by testing them against given sequence values first. This catches formula misreading errors before you invest time in calculations with the wrong pattern.

Question 13

A geometric sequence has its first term equal to 3 and its fourth term equal to 192. What is the second term?

  1. 12 (correct answer)
  2. 18
  3. 24
  4. 48
Explanation: When you encounter a geometric sequence problem, remember that each term is found by multiplying the previous term by a constant ratio. Your goal is to find this common ratio first, then work systematically through the sequence. In a geometric sequence, if the first term is aa and the common ratio is rr, then the fourth term equals a⋅r3a \cdot r^3. Here, you know the first term is 3 and the fourth term is 192, so: 3⋅r3=1923 \cdot r^3 = 192 Solving for the ratio: r3=64r^3 = 64, which means r=4r = 4. Now you can find the second term: 3×4=123 \times 4 = 12. Let's verify by checking the complete sequence: 3, 12, 48, 192. Each term is indeed 4 times the previous term, and the fourth term is 192 as required. Looking at the wrong answers: (B) 18 would give a ratio of 6, making the sequence 3, 18, 108, 648 – the fourth term would be far too large. (C) 24 creates a ratio of 8, yielding 3, 24, 192, 1536 – here the third term already equals our target fourth term. (D) 48 means a ratio of 16, producing 3, 48, 768, 12288 – again, the fourth term is much too large. The key strategy for geometric sequences is always to find the common ratio first using the given terms, then build the sequence step by step. Don't try to guess – let the algebra guide you to the unique correct ratio.

Question 14

In the sequence 2,6,18,54,162,...2, 6, 18, 54, 162, ... each term is obtained by multiplying the previous term by 3. If this pattern continues, which term will first exceed 10,000?

  1. 8th term
  2. 9th term (correct answer)
  3. 10th term
  4. 11th term
Explanation: This question tests your understanding of geometric sequences, where each term is found by multiplying the previous term by a constant ratio. When you see a pattern like this, you need to find the general formula and then determine which term reaches the target value. In this sequence, each term equals the previous term times 3, so the common ratio is 3. The general formula for the nth term is: an=2×3n−1a_n = 2 \times 3^{n-1}, where 2 is the first term. To find when this first exceeds 10,000, you need to solve: 2×3n−1>10,0002 \times 3^{n-1} > 10,000, which gives us 3n−1>5,0003^{n-1} > 5,000. Let's calculate the terms systematically: 8th term = 2×37=2×2,187=4,3742 \times 3^7 = 2 \times 2,187 = 4,374, and 9th term = 2×38=2×6,561=13,1222 \times 3^8 = 2 \times 6,561 = 13,122. Since 13,122 > 10,000, the 9th term is the first to exceed 10,000. Choice A (8th term) falls short because 4,374 < 10,000. Choice C (10th term) would be correct if the question asked for the second term to exceed 10,000, since that equals 39,366. Choice D (11th term) represents an even later term that's unnecessary since we find our answer earlier. When working with geometric sequences, always write out the general formula first, then calculate terms systematically rather than trying to guess. This prevents calculation errors and helps you see the pattern clearly.

Question 15

A sequence follows the pattern where the nnth term is given by an=(−1)nn+1a_n = \frac{(-1)^n}{n+1}. What is the sum of the 5th and 6th terms?

  1. −142-\frac{1}{42} (correct answer)
  2. 142\frac{1}{42}
  3. −121-\frac{1}{21}
  4. 121\frac{1}{21}
Explanation: When you encounter a sequence with an alternating sign pattern like (−1)n(-1)^n, you're dealing with terms that switch between positive and negative based on whether nn is even or odd. To find the sum of the 5th and 6th terms, calculate each term using the formula an=(−1)nn+1a_n = \frac{(-1)^n}{n+1}. For the 5th term: a5=(−1)55+1=−16=−16a_5 = \frac{(-1)^5}{5+1} = \frac{-1}{6} = -\frac{1}{6} For the 6th term: a6=(−1)66+1=17a_6 = \frac{(-1)^6}{6+1} = \frac{1}{7} Now add them: a5+a6=−16+17a_5 + a_6 = -\frac{1}{6} + \frac{1}{7} To add these fractions, find a common denominator: −16+17=−742+642=−142-\frac{1}{6} + \frac{1}{7} = -\frac{7}{42} + \frac{6}{42} = -\frac{1}{42} This confirms answer A is correct. Answer B (142\frac{1}{42}) results from a sign error—likely forgetting that (−1)5=−1(-1)^5 = -1, making the 5th term positive instead of negative. Answer C (−121-\frac{1}{21}) comes from incorrectly using 21 as the common denominator instead of 42, or miscalculating 6×76 \times 7. Answer D (121\frac{1}{21}) combines both the sign error and the denominator mistake. Remember that (−1)n(-1)^n equals −1-1 when nn is odd and +1+1 when nn is even. Always double-check your arithmetic when adding fractions—find the least common multiple of the denominators carefully.

Question 16

The sequence 1,1,2,3,5,8,13,...1, 1, 2, 3, 5, 8, 13, ... follows the Fibonacci pattern where each term after the second is the sum of the two preceding terms. What is the 12th term?

  1. 89
  2. 144 (correct answer)
  3. 233
  4. 377
Explanation: When you encounter a Fibonacci sequence problem, you need to systematically calculate each term using the defining rule: each term equals the sum of the two preceding terms. Starting with the given sequence 1,1,2,3,5,8,13,...1, 1, 2, 3, 5, 8, 13, ..., let's continue calculating to find the 12th term:
  • 1st term: 1
  • 2nd term: 1
  • 3rd term: 1+1=21 + 1 = 2
  • 4th term: 1+2=31 + 2 = 3
  • 5th term: 2+3=52 + 3 = 5
  • 6th term: 3+5=83 + 5 = 8
  • 7th term: 5+8=135 + 8 = 13
  • 8th term: 8+13=218 + 13 = 21
  • 9th term: 13+21=3413 + 21 = 34
  • 10th term: 21+34=5521 + 34 = 55
  • 11th term: 34+55=8934 + 55 = 89
  • 12th term: 55+89=14455 + 89 = 144
The 12th term is 144, making (B) correct. Looking at the wrong answers: (A) 89 is actually the 11th term—a common error if you miscount positions or stop calculating one step too early. (C) 233 would be the 13th term (89+144=23389 + 144 = 233), showing what happens when you go one step too far. (D) 377 is the 14th term, indicating even more overcalculation. Strategy tip: For Fibonacci problems, always write out each term systematically and double-check your position counting. The most common errors come from miscounting which term you're looking for or making arithmetic mistakes in the addition sequence.

Question 17

A geometric sequence has the property that the ratio of any term to the previous term is constant. If the 2nd term is 6 and the 5th term is 162, what is the 7th term?

  1. 1458 (correct answer)
  2. 1944
  3. 2187
  4. 2916
Explanation: Geometric sequences are foundational in quantitative reasoning, where each term equals the previous term multiplied by a constant ratio (r). When you see a geometric sequence problem, your goal is to find this common ratio first, then use it to calculate unknown terms. To find the common ratio, use the relationship between the given terms. The 2nd term is 6 and the 5th term is 162. Since each term multiplies by r, the 5th term equals the 2nd term times r3r^3 (three steps from 2nd to 5th). So: 162=6×r3162 = 6 \times r^3, which gives us r3=27r^3 = 27, therefore r=3r = 3. Now you can find the 7th term. From the 5th term (162) to the 7th term requires multiplying by r2r^2: 162×32=162×9=1458162 \times 3^2 = 162 \times 9 = 1458. Let's examine why each wrong answer appears: (B) 1944 might result from incorrectly calculating 162×12162 \times 12 if you mistakenly thought the ratio was 6 instead of 3. (C) 2187 equals 373^7, which could tempt you if you confused the term position with the actual term value. (D) 2916 equals 162×18162 \times 18, suggesting an error where you might have used 32×2=183^2 \times 2 = 18 instead of just 32=93^2 = 9. The correct answer is (A) 1458. Strategy tip: Always find the common ratio first by setting up the relationship between known terms, then systematically multiply to reach your target term. Double-check by ensuring your ratio is consistent throughout the sequence.

Question 18

In an arithmetic sequence, the 3rd term is 17 and the 8th term is 42. What is the 15th term?

  1. 77 (correct answer)
  2. 79
  3. 81
  4. 83
Explanation: When you encounter arithmetic sequence problems, you're working with patterns where each term increases by the same constant difference. The key is finding that common difference and using the arithmetic sequence formula. In an arithmetic sequence, any term can be found using an=a1+(n−1)da_n = a_1 + (n-1)d, where a1a_1 is the first term and dd is the common difference. Since you know the 3rd term is 17 and the 8th term is 42, you can set up equations: a3=a1+2d=17a_3 = a_1 + 2d = 17 and a8=a1+7d=42a_8 = a_1 + 7d = 42. To find the common difference, subtract the first equation from the second: (a1+7d)−(a1+2d)=42−17(a_1 + 7d) - (a_1 + 2d) = 42 - 17, which gives you 5d=255d = 25, so d=5d = 5. Now find the first term by substituting back: a1+2(5)=17a_1 + 2(5) = 17, so a1=7a_1 = 7. For the 15th term: a15=7+(15−1)(5)=7+70=77a_{15} = 7 + (15-1)(5) = 7 + 70 = 77. Looking at the wrong answers: B) 79 results from miscalculating the common difference as 6 instead of 5. C) 81 comes from incorrectly using d=5d = 5 but making an arithmetic error in the final calculation. D) 83 typically results from finding the wrong first term, perhaps using a1=9a_1 = 9 instead of 7. Remember this pattern: when given two terms in an arithmetic sequence, always subtract to find the common difference first. The difference between the terms divided by the difference between their positions gives you dd.

Question 19

A sequence begins with 1, 3, 6, 10, 15, ... where each term represents triangular numbers (sums of consecutive integers starting from 1). What is the 12th triangular number?

  1. 66
  2. 78 (correct answer)
  3. 91
  4. 105
Explanation: When you encounter a sequence involving "triangular numbers," you're dealing with a specific pattern where each term represents the sum of consecutive positive integers starting from 1. The sequence 1, 3, 6, 10, 15... follows this pattern: 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5... To find the 12th triangular number, you need the sum 1+2+3+...+121+2+3+...+12. Rather than adding all these numbers individually, use the triangular number formula: Tn=n(n+1)2T_n = \frac{n(n+1)}{2}, where nn is the position in the sequence. For the 12th triangular number: T12=12(12+1)2=12×132=1562=78T_{12} = \frac{12(12+1)}{2} = \frac{12 \times 13}{2} = \frac{156}{2} = 78. This confirms that B) 78 is correct. Looking at the wrong answers: A) 66 would be the result if you mistakenly used n=11n = 11 instead of 12, giving you 11×122=66\frac{11 \times 12}{2} = 66. C) 91 comes from using n=13n = 13, calculating 13×142=91\frac{13 \times 14}{2} = 91. D) 105 results from using n=14n = 14, giving 14×152=105\frac{14 \times 15}{2} = 105. These are all "off-by-one" or "off-by-two" errors that occur when you miscount the position. Remember the triangular number formula Tn=n(n+1)2T_n = \frac{n(n+1)}{2} — it's much faster than adding consecutive integers manually. Always double-check that you're using the correct value of nn for the position you want.

Question 20

In an arithmetic sequence, the sum of the 3rd and 7th terms is 40, and the sum of the 5th and 9th terms is 56. What is the 10th term?

  1. 34
  2. 36
  3. 38
  4. 40 (correct answer)
Explanation: Arithmetic sequences are foundational in algebra, where each term increases by the same constant difference. When you see problems involving multiple terms and their sums, you'll need to set up equations using the general formula for the nth term: an=a1+(n−1)da_n = a_1 + (n-1)d, where a1a_1 is the first term and dd is the common difference. Let's work with what we know. The 3rd term is a3=a1+2da_3 = a_1 + 2d and the 7th term is a7=a1+6da_7 = a_1 + 6d. Since their sum is 40: a3+a7=(a1+2d)+(a1+6d)=2a1+8d=40a_3 + a_7 = (a_1 + 2d) + (a_1 + 6d) = 2a_1 + 8d = 40. Similarly, a5+a9=(a1+4d)+(a1+8d)=2a1+12d=56a_5 + a_9 = (a_1 + 4d) + (a_1 + 8d) = 2a_1 + 12d = 56. Now you have two equations: 2a1+8d=402a_1 + 8d = 40 and 2a1+12d=562a_1 + 12d = 56. Subtracting the first from the second: 4d=164d = 16, so d=4d = 4. Substituting back: 2a1+8(4)=402a_1 + 8(4) = 40, which gives a1=4a_1 = 4. The 10th term is a10=4+9(4)=40a_{10} = 4 + 9(4) = 40. Choice A (34) would result from miscalculating the common difference as 3 instead of 4. Choice B (36) occurs if you confuse the 9th term for the 10th. Choice C (38) comes from an arithmetic error when finding a1a_1, likely calculating it as 2 instead of 4. Remember: when working with arithmetic sequences, always double-check your system of equations and verify your common difference by substituting back into the original conditions.