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

The second, third, and fourth terms of a geometric sequence are 108, 72, and 48, respectively. What is the first term of the sequence?

162
144
81
72
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ISEE Middle Level Quiz

ISEE Middle Level Quiz: Missing Terms In Sequences

Practice Missing Terms In Sequences in ISEE Middle 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 Middle 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

The second, third, and fourth terms of a geometric sequence are 108, 72, and 48, respectively. What is the first term of the sequence?

  1. 162 (correct answer)
  2. 144
  3. 81
  4. 72
Explanation: First, find the common ratio rr by dividing a term by its preceding term: r=72÷108=23r = 72 \div 108 = \frac{2}{3}. To find the first term, we must reverse the process. Instead of multiplying by the ratio to get the next term, we divide the second term by the ratio to get the first term. 108÷23=108×32=162108 \div \frac{2}{3} = 108 \times \frac{3}{2} = 162.

Question 2

A special sequence is formed by the rule that each term after the second is the difference of the two preceding terms (the previous term subtracted from the one before it). What is the next term in this sequence: 20, 12, 8, 4, 4, ...?

  1. 0 (correct answer)
  2. 4
  3. 8
  4. -4
Explanation: The rule is to subtract the second of two consecutive terms from the first to get the next term. Let's check the given sequence: 20−12=820 - 12 = 8; 12−8=412 - 8 = 4; 8−4=48 - 4 = 4. To find the next term, we apply the rule to the last two terms: 4−4=04 - 4 = 0.

Question 3

A baker decorates a cake with concentric rings of frosting dots. The first, innermost ring has 8 dots. Each successive ring going outwards has 6 more dots than the one before it. How many dots are in the 7th ring?

  1. 38
  2. 42
  3. 44 (correct answer)
  4. 50
Explanation: This describes an arithmetic sequence where the first term is a1=8a_1 = 8 and the common difference is d=6d = 6. We need to find the 7th term, a7a_7. Using the formula an=a1+(n−1)da_n = a_1 + (n-1)d, we get a7=8+(7−1)×6=8+6×6=8+36=44a_7 = 8 + (7-1) \times 6 = 8 + 6 \times 6 = 8 + 36 = 44. So, there are 44 dots in the 7th ring.

Question 4

Identify the missing term in the pattern: 8, 11, 14, ‾\underline{\hspace{2em}}.

  1. 15
  2. 16
  3. 17 (correct answer)
  4. 18
Explanation: This question tests middle school mathematics achievement, specifically finding missing terms in a sequence. Understanding sequences involves recognizing patterns such as arithmetic (adding a constant) and geometric (multiplying by a constant). These patterns allow us to determine missing terms by applying consistent rules. In the provided sequence 8, 11, 14, __, the pattern involves adding 3 to each term (11-8=3, 14-11=3). The correct answer is C (17) because it logically follows the pattern: 8+3=11, 11+3=14, 14+3=17. Choice A (15) is incorrect because it assumes adding only 1, while choice D (18) assumes adding 4.

Question 5

What is the next number in the following pattern? 5, 3, 9, 7, 21, 19, ...

  1. 17
  2. 38
  3. 51
  4. 57 (correct answer)
Explanation: The pattern alternates between two operations: subtracting 2 and multiplying by 3. 5−2=35 - 2 = 3, 3×3=93 \times 3 = 9, 9−2=79 - 2 = 7, 7×3=217 \times 3 = 21, 21−2=1921 - 2 = 19. The next operation is to multiply by 3: 19×3=5719 \times 3 = 57.

Question 6

Find the next term in the sequence: 3, 7, 14, 24, 37, ...

  1. 49
  2. 50
  3. 53 (correct answer)
  4. 54
Explanation: This sequence does not have a common difference. Instead, the differences between consecutive terms form a pattern. The differences are 7−3=47-3=4, 14−7=714-7=7, 24−14=1024-14=10, 37−24=1337-24=13. This sequence of differences (4, 7, 10, 13) is an arithmetic sequence with a common difference of 3. The next difference will be 13+3=1613+3=16. Therefore, the next term in the original sequence is 37+16=5337+16=53.

Question 7

Find the missing term in the sequence: 4, 1, 8, 4, 16, 7, 32, ‾\underline{\hspace{2em}}.

  1. 9
  2. 10 (correct answer)
  3. 28
  4. 64
Explanation: This sequence is composed of two interleaved sequences. The first sequence, in the odd positions, is 4, 8, 16, 32, ... (each term is multiplied by 2). The second sequence, in the even positions, is 1, 4, 7, ... (each term has 3 added to it). The missing term is the eighth term, so it belongs to the second sequence. The next term in that sequence is 7+3=107 + 3 = 10.

Question 8

Which number completes the sequence: 5, 9, ‾\underline{\hspace{2em}}, 17?

  1. 11
  2. 12
  3. 13 (correct answer)
  4. 14
Explanation: This question tests middle school mathematics achievement, specifically finding missing terms in a sequence. Understanding sequences involves recognizing patterns such as arithmetic (adding a constant) and geometric (multiplying by a constant). These patterns allow us to determine missing terms by applying consistent rules. In the provided sequence 5, 9, __, 17, the pattern involves adding 4 to each term (9-5=4, and we need to verify 17-13=4). The correct answer is C (13) because it logically follows the pattern: 5+4=9, 9+4=13, 13+4=17. Choice A (11) is incorrect because it assumes adding only 2, while choice D (14) assumes adding 5, both representing common errors in pattern recognition.

Question 9

Find the number that completes the sequence: 0, 7, 26, 63, 124, ‾\underline{\hspace{2em}}.

  1. 185
  2. 205
  3. 215 (correct answer)
  4. 216
Explanation: The pattern for this sequence is n3−1n^3 - 1, where n is the term's position in the sequence (starting with n=1). The first term is 13−1=01^3 - 1 = 0. The second is 23−1=8−1=72^3 - 1 = 8 - 1 = 7. The third is 33−1=27−1=263^3 - 1 = 27 - 1 = 26. The fourth is 43−1=64−1=634^3 - 1 = 64 - 1 = 63. The fifth is 53−1=125−1=1245^3 - 1 = 125 - 1 = 124. The next term will be the sixth term: 63−1=216−1=2156^3 - 1 = 216 - 1 = 215.

Question 10

What is the next term in the sequence 4, 11, 25, 53, ...?

  1. 81
  2. 92
  3. 106
  4. 109 (correct answer)
Explanation: The pattern in this sequence is to multiply the previous term by 2 and then add 3. Let's check: 4×2+3=8+3=114 \times 2 + 3 = 8 + 3 = 11. 11×2+3=22+3=2511 \times 2 + 3 = 22 + 3 = 25. 25×2+3=50+3=5325 \times 2 + 3 = 50 + 3 = 53. To find the next term, we apply the rule to 53: 53×2+3=106+3=10953 \times 2 + 3 = 106 + 3 = 109.

Question 11

What term comes next in the sequence 2, 5, 10, 17, 26, ...?

  1. 35
  2. 36
  3. 37 (correct answer)
  4. 38
Explanation: The pattern for this sequence is n2+1n^2 + 1, where n is the term's position in the sequence (starting with n=1). The first term is 12+1=21^2 + 1 = 2. The second term is 22+1=52^2 + 1 = 5. The third term is 32+1=103^2 + 1 = 10. The fourth term is 42+1=174^2 + 1 = 17. The fifth term is 52+1=265^2 + 1 = 26. The next term (the sixth term) is 62+1=36+1=376^2 + 1 = 36 + 1 = 37.

Question 12

What is the next term in the pattern: 100, 99, 96, 91, 84, ...?

  1. 77
  2. 76
  3. 75 (correct answer)
  4. 73
Explanation: The differences between consecutive terms are increasing. The differences are: 100−99=1100 - 99 = 1, 99−96=399 - 96 = 3, 96−91=596 - 91 = 5, 91−84=791 - 84 = 7. The amounts being subtracted are consecutive odd numbers. The next odd number to subtract is 9. Therefore, the next term is 84−9=7584 - 9 = 75.

Question 13

What is the next number in the following sequence? 10, 20, 15, 25, 20, 30, ...

  1. 20
  2. 25 (correct answer)
  3. 35
  4. 40
Explanation: The sequence follows a repeating pattern of two operations: add 10, then subtract 5. 10+10=2010 + 10 = 20; 20−5=1520 - 5 = 15; 15+10=2515 + 10 = 25; 25−5=2025 - 5 = 20; 20+10=3020 + 10 = 30. The next step is to subtract 5: 30−5=2530 - 5 = 25.

Question 14

If the pattern continues, what is the next fraction in the sequence? 12,35,58,711,...\frac{1}{2}, \frac{3}{5}, \frac{5}{8}, \frac{7}{11}, ...

  1. 814\frac{8}{14}
  2. 913\frac{9}{13}
  3. 914\frac{9}{14} (correct answer)
  4. 1114\frac{11}{14}
Explanation: There are two separate patterns for the numerators and the denominators. The numerators are 1, 3, 5, 7, ... which are consecutive odd numbers. The next numerator will be 9. The denominators are 2, 5, 8, 11, ... which form an arithmetic sequence with a common difference of 3. The next denominator will be 11+3=1411 + 3 = 14. Therefore, the next fraction in the sequence is 914\frac{9}{14}.

Question 15

What is the missing term in the sequence: 30, 25, ‾\underline{\hspace{2em}}, 15?

  1. 18
  2. 19
  3. 20 (correct answer)
  4. 21
Explanation: This question tests middle school mathematics achievement, specifically finding missing terms in a sequence. Understanding sequences involves recognizing patterns such as arithmetic (adding a constant) and geometric (multiplying by a constant). These patterns allow us to determine missing terms by applying consistent rules. In the provided sequence 30, 25, __, 15, the pattern involves subtracting 5 from each term (30-5=25, and we need to verify with 15). The correct answer is C (20) because it logically follows the pattern: 30-5=25, 25-5=20, 20-5=15. Choice A (18) is incorrect because it assumes subtracting 7, while choice B (19) assumes subtracting 6.

Question 16

The first three terms of an arithmetic sequence are 5, 9, and 13. What is the 10th term in the sequence?

  1. 37
  2. 41 (correct answer)
  3. 45
  4. 36
Explanation: The sequence is an arithmetic sequence with a first term a1=5a_1 = 5. The common difference dd is 9−5=49 - 5 = 4. The formula for the nth term of an arithmetic sequence is an=a1+(n−1)da_n = a_1 + (n-1)d. To find the 10th term (a10a_{10}), we substitute the values: a10=5+(10−1)×4=5+9×4=5+36=41a_{10} = 5 + (10-1) \times 4 = 5 + 9 \times 4 = 5 + 36 = 41.

Question 17

Find the next number in the sequence: 12, 18, 27, 40.5, ...

  1. 54
  2. 55.5
  3. 60.5
  4. 60.75 (correct answer)
Explanation: This is a geometric sequence. To find the common ratio, divide a term by its preceding term: 18÷12=1.518 \div 12 = 1.5. Let's check with another pair: 27÷18=1.527 \div 18 = 1.5. The common ratio is 1.5. To find the next term, multiply the last term by 1.5: 40.5×1.5=60.7540.5 \times 1.5 = 60.75.

Question 18

Find the next number in the pattern: 5, 6, 10, 19, 35, ...

  1. 51
  2. 55
  3. 60 (correct answer)
  4. 61
Explanation: Let's examine the differences between consecutive terms: 6−5=16-5=1, 10−6=410-6=4, 19−10=919-10=9, 35−19=1635-19=16. The differences are 1, 4, 9, 16, which are the perfect squares 12,22,32,421^2, 2^2, 3^2, 4^2. The next difference in the pattern will be 52=255^2 = 25. Adding this to the last term gives the next term in the sequence: 35+25=6035 + 25 = 60.

Question 19

What is the next number in the sequence? 2, 5, 7, 12, 19, ...

  1. 26
  2. 29
  3. 31 (correct answer)
  4. 38
Explanation: This is a Fibonacci-style sequence, where each term is the sum of the two preceding terms. 2+5=72 + 5 = 7, 5+7=125 + 7 = 12, 7+12=197 + 12 = 19. The next term is found by adding the last two terms: 12+19=3112 + 19 = 31.

Question 20

Find the next term in the sequence: 400, 100, 25, 6.25, ...

  1. 1.5625 (correct answer)
  2. -12.5
  3. 1.25
  4. 5.0
Explanation: This is a geometric sequence where each term is found by dividing the previous term by a constant number. 400÷4=100400 \div 4 = 100; 100÷4=25100 \div 4 = 25; 25÷4=6.2525 \div 4 = 6.25. To find the next term, we divide 6.25 by 4: 6.25÷4=1.56256.25 \div 4 = 1.5625.