Missing Sequence Terms - ISEE Middle Level Quantitative Reasoning

What is the missing term in this sequence: $6, 10, 14, ?, 22$ ?
Answer: 18 This question tests middle school quantitative reasoning skills, specifically finding a missing term in a sequence. Understanding sequences involves recognizing patterns such as arithmetic or geometric progressions. In this sequence (6, 10, 14, ?, 22), each term follows the rule of adding 4 to the previous term: 6+4=10, 10+4=14, 14+4=18, 18+4=22. The correct answer, 18 (C), works because it follows the pattern, ensuring sequential consistency. A common distractor might suggest 16 (A), which fails because it only adds 2 instead of 4. To improve, students should practice identifying sequence types and applying the correct operations, ensuring they double-check their calculations and rule applications.
Following the pattern described, how many bacteria will be in the culture on Day 5?
Answer: 90 The net change each day is an increase of 15 and a decrease of 5, which is a net increase of 10 bacteria per day. This is an arithmetic sequence with a common difference of 10. Day 1: 50 Day 2: 50 + 10 = 60 Day 3: 60 + 10 = 70 Day 4: 70 + 10 = 80 Day 5: 80 + 10 = 90 There will be 90 bacteria on Day 5.
A sequence follows the pattern: add 8, then subtract 3. If the first term is 5, what is the 6th term of the sequence?
Answer: 23 The sequence starts at 5 and alternates between adding 8 and subtracting 3. Term 1: 5, Term 2: 5 + 8 = 13, Term 3: 13 - 3 = 10, Term 4: 10 + 8 = 18, Term 5: 18 - 3 = 15, Term 6: 15 + 8 = 23. The 6th term is 23.
What is the missing term: 11, 14, 17, __, 23?
Answer: 20 This question tests middle school quantitative reasoning skills, specifically finding a missing term in a sequence. Understanding sequences involves recognizing patterns such as arithmetic or geometric progressions. In this sequence, each term follows the rule of adding 3 to the previous term. The correct answer, 20, works because it follows the pattern, ensuring sequential consistency by adding 3 to 17 and leading to 23 by adding another 3. A common distractor might suggest 18, which fails because it assumes addition of 1 instead of 3. To improve, students should practice identifying sequence types and applying the correct operations, ensuring they double-check their calculations and rule applications.
What is the missing term: 5, 15, 45, __, 405?
Answer: 135 This question tests middle school quantitative reasoning skills, specifically finding a missing term in a sequence. Understanding sequences involves recognizing patterns such as arithmetic or geometric progressions. In this sequence, each term follows the rule of multiplying by 3. The correct answer, 135, works because it follows the pattern, ensuring sequential consistency by multiplying 45 by 3 and leading to 405 by multiplying again by 3. A common distractor might suggest 120, which fails because it assumes an incorrect multiplier like 2.666. To improve, students should practice identifying sequence types and applying the correct operations, ensuring they double-check their calculations and rule applications.
A recursive sequence is defined by a(1) = 2, and a(n) = a(n-1) + n + 1 for n > 1. What is the value of a(4)?
Answer: 14 We are given the first term and a rule to find the subsequent terms. a(1) = 2 To find a(2), let n=2: a(2) = a(1) + 2 + 1 = 2 + 3 = 5. To find a(3), let n=3: a(3) = a(2) + 3 + 1 = 5 + 4 = 9. To find a(4), let n=4: a(4) = a(3) + 4 + 1 = 9 + 5 = 14.
Fill in the blank: 1, 4, 16, __, 256.
Answer: 64 This question tests middle school quantitative reasoning skills, specifically finding a missing term in a sequence. Understanding sequences involves recognizing patterns such as arithmetic or geometric progressions. In this sequence, each term follows the rule of multiplying by 4. The correct answer, 64, works because it follows the pattern, ensuring sequential consistency by multiplying 16 by 4 and leading to 256 by multiplying again by 4. A common distractor might suggest 48, which fails because it assumes multiplication by 3. To improve, students should practice identifying sequence types and applying the correct operations, ensuring they double-check their calculations and rule applications.
Consider the sequence where the n-th term is given by the formula 3n² - 5. What is the 6th term of this sequence?
Answer: 103 To find the 6th term, substitute n = 6 into the formula 3n² - 5. Term 6 = 3 * (6²) - 5 Term 6 = 3 * 36 - 5 Term 6 = 108 - 5 Term 6 = 103
What is the missing term: 4, 12, 36, __, 324?
Answer: 108 This question tests middle school quantitative reasoning skills, specifically finding a missing term in a sequence. Understanding sequences involves recognizing patterns such as arithmetic or geometric progressions. In this sequence, each term follows the rule of multiplying by 3. The correct answer, 108, works because it follows the pattern, ensuring sequential consistency by multiplying 36 by 3 and leading to 324 by multiplying again by 3. A common distractor might suggest 96, which fails because it assumes multiplication by about 2.666. To improve, students should practice identifying sequence types and applying the correct operations, ensuring they double-check their calculations and rule applications.
In the sequence 5, 6, 8, 11, 15, ..., the difference between consecutive terms increases by 1 each time. If this pattern continues, what is the 8th term of the sequence?
Answer: 33 The sequence is 5, 6, 8, 11, 15, ... The differences are: 6 - 5 = 1 8 - 6 = 2 11 - 8 = 3 15 - 11 = 4 The pattern of the differences is +1, +2, +3, +4, ... To find the next terms: The 6th term = 15 + 5 = 20. The 7th term = 20 + 6 = 26. The 8th term = 26 + 7 = 33.
Complete the sequence: 30, 25, 20, __, 10.
Answer: 15 This question tests middle school quantitative reasoning skills, specifically finding a missing term in a sequence. Understanding sequences involves recognizing patterns such as arithmetic or geometric progressions. In this sequence, each term follows the rule of subtracting 5 from the previous term. The correct answer, 15, works because it follows the pattern, ensuring sequential consistency by subtracting 5 from 20 and leading to 10 by subtracting another 5. A common distractor might suggest 12, which fails because it assumes a subtraction of 8 instead of 5. To improve, students should practice identifying sequence types and applying the correct operations, ensuring they double-check their calculations and rule applications.
Consider the following sequence: 3, 4, 7, 11, 18, 29, ... What is the first term in the sequence that is greater than 100?
Answer: 123 This is a Fibonacci-style sequence where each term is the sum of the two preceding terms: 7 = 3+4, 11 = 4+7, 18 = 7+11, 29 = 11+18. Continuing the pattern: 29 + 18 = 47, 47 + 29 = 76, 76 + 47 = 123. The first term greater than 100 is 123.
Fill in the blank: $3, 6, 12, ?, 48$ .
Answer: 24 This question tests middle school quantitative reasoning skills, specifically finding a missing term in a sequence. Understanding sequences involves recognizing patterns such as arithmetic or geometric progressions. In this sequence (3, 6, 12, ?, 48), each term follows the rule of multiplying by 2: 3×2=6, 6×2=12, 12×2=24, 24×2=48. The correct answer, 24 (C), works because it follows the pattern, ensuring sequential consistency. A common distractor might suggest 18 (A), which fails because it adds 6 instead of multiplying by 2. To improve, students should practice identifying sequence types and applying the correct operations, ensuring they double-check their calculations and rule applications.
Consider the sequence: 4, 12, 9, 27, 24, ... Which rule describes the pattern?
Answer: Multiply by 3, subtract 3 When you encounter a sequence problem, you need to identify the pattern by examining how each term relates to the previous one. Look at the transitions between consecutive terms to find the rule. Let's trace through this sequence: 4, 12, 9, 27, 24, ... From 4 to 12: $4 \times 3 = 12$ From 12 to 9: $12 - 3 = 9$ From 9 to 27: $9 \times 3 = 27$ From 27 to 24: $27 - 3 = 24$ The pattern alternates: multiply by 3, then subtract 3, then multiply by 3, then subtract 3. This confirms that choice D) "Multiply by 3, subtract 3" correctly describes the repeating pattern. Let's check why the other options fail. Choice A) "Multiply by 3, add 3" works for the first step ( $4 \times 3 = 12$ ), but then $12 + 3 = 15$ , not 9. Choice B) "Add 8, subtract 3" gives us $4 + 8 = 12$ for the first step, but then $12 - 3 = 9$ works, yet $9 + 8 = 17$ , not 27. Choice C) "Add 8, divide by 4/3" starts correctly with $4 + 8 = 12$ , but $12 \div \frac{4}{3} = 12 \times \frac{3}{4} = 9$ works, then $9 + 8 = 17$ , which doesn't equal 27. For sequence problems on the ISEE, always test your suspected pattern on at least three transitions to confirm it's consistent. Many wrong answers will work for just one or two steps but break down when you check further.
The first term of a sequence is 800. Each subsequent term is found by dividing the previous term by 2. What is the first term in the sequence that is less than 20?
Answer: 12.5 The sequence starts with 800 and each term is half of the previous one. Term 1: 800 Term 2: 800 / 2 = 400 Term 3: 400 / 2 = 200 Term 4: 200 / 2 = 100 Term 5: 100 / 2 = 50 Term 6: 50 / 2 = 25 Term 7: 25 / 2 = 12.5 Term 6 is 25, which is not less than 20. Term 7 is 12.5, which is the first term less than 20.
Identify the missing term: $2, 3, 5, 8, ?, 21$ .
Answer: 13 This question tests middle school quantitative reasoning skills, specifically finding a missing term in a sequence. Understanding sequences involves recognizing patterns such as arithmetic or geometric progressions. In this sequence (2, 3, 5, 8, ?, 21), each term follows the Fibonacci-like rule where each term is the sum of the two preceding terms: 2+3=5, 3+5=8, 5+8=13, 8+13=21. The correct answer, 13 (C), works because it follows the pattern, ensuring sequential consistency. A common distractor might suggest 11 (A), which fails because it doesn't follow the sum rule. To improve, students should practice identifying sequence types and applying the correct operations, ensuring they double-check their calculations and rule applications.
A sequence is generated by taking the previous term, multiplying by -1, and adding 2. If the first term is 5, what is the sum of the first four terms?
Answer: 4 First, we need to find the first four terms of the sequence. Term 1 = 5 (given) Term 2 = (5 * -1) + 2 = -5 + 2 = -3 Term 3 = (-3 * -1) + 2 = 3 + 2 = 5 Term 4 = (5 * -1) + 2 = -5 + 2 = -3 The sequence is 5, -3, 5, -3, ... Now, we find the sum of the first four terms: 5 + (-3) + 5 + (-3) = 2 + 2 = 4.
The terms in a sequence are 1/2, 1, 2, 4, ... If the pattern continues, what is the 10th term?
Answer: 256 This is a geometric sequence where each term is found by multiplying the previous term by 2. The first term is 1/2. The general form is a* $r^{n-1}$ . Here a = 1/2 and r = 2. The 10th term is (1/2) * $2^{10-1}$ = (1/2) * $2^9$ = $2^8$ . $2^1$ = 2, $2^2$ = 4, $2^3$ = 8, $2^4$ = 16, $2^5$ = 32, $2^6$ = 64, $2^7$ = 128, $2^8$ = 256. Alternatively, list the terms: T1=1/2, T2=1, T3=2, T4=4, T5=8, T6=16, T7=32, T8=64, T9=128, T10=256.
Complete the sequence: 100, 50, 25, __, $\tfrac{25}{4}$ .
Answer: 12.5 This question tests middle school quantitative reasoning skills, specifically finding a missing term in a sequence. Understanding sequences involves recognizing patterns such as arithmetic or geometric progressions. In this sequence, each term follows the rule of dividing by 2. The correct answer, 12.5, works because it follows the pattern, ensuring sequential consistency by dividing 25 by 2 and leading to 25/4 by dividing again by 2. A common distractor might suggest 10, which fails because it assumes subtraction instead of division. To improve, students should practice identifying sequence types and applying the correct operations, ensuring they double-check their calculations and rule applications.
A sequence starts with 1. Each subsequent term is found by doubling the previous term and adding the new term's position number. What is the 5th term?
Answer: 57 Following the rule: Term 1 = 1, Term 2 = (2×1) + 2 = 4, Term 3 = (2×4) + 3 = 11, Term 4 = (2×11) + 4 = 26, Term 5 = (2×26) + 5 = 57. The 5th term is 57.

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