ISEE Middle Level Mathematics Achievement Quiz: Missing Terms In Sequences

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 Mathematics Achievement.

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.

Question 1

1. 162 (correct answer)
2. 144
3. 81
4. 72

Explanation: First, find the common ratio $r$ by dividing a term by its preceding term: $r = 72 \div 108 = \frac{2}{3}$. 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. $108 \div \frac{2}{3} = 108 \times \frac{3}{2} = 162$.

Question 2

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: $20 - 12 = 8$; $12 - 8 = 4$; $8 - 4 = 4$. To find the next term, we apply the rule to the last two terms: $4 - 4 = 0$.

Question 3

1. 38
2. 42
3. 44 (correct answer)
4. 50

Explanation: This describes an arithmetic sequence where the first term is $a_1 = 8$ and the common difference is $d = 6$. We need to find the 7th term, $a_7$. Using the formula $a_n = a_1 + (n-1)d$, we get $a_7 = 8 + (7-1) \times 6 = 8 + 6 \times 6 = 8 + 36 = 44$. So, there are 44 dots in the 7th ring.

Question 4

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

1. 17
2. 38
3. 51
4. 57 (correct answer)

Explanation: The pattern alternates between two operations: subtracting 2 and multiplying by 3. $5 - 2 = 3$, $3 \times 3 = 9$, $9 - 2 = 7$, $7 \times 3 = 21$, $21 - 2 = 19$. The next operation is to multiply by 3: $19 \times 3 = 57$.

Question 6

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 $7-3=4$, $14-7=7$, $24-14=10$, $37-24=13$. This sequence of differences (4, 7, 10, 13) is an arithmetic sequence with a common difference of 3. The next difference will be $13+3=16$. Therefore, the next term in the original sequence is $37+16=53$.

Question 7

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 $7 + 3 = 10$.

Question 8

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

1. 185
2. 205
3. 215 (correct answer)
4. 216

Explanation: The pattern for this sequence is $n^3 - 1$, where n is the term's position in the sequence (starting with n=1). The first term is $1^3 - 1 = 0$. The second is $2^3 - 1 = 8 - 1 = 7$. The third is $3^3 - 1 = 27 - 1 = 26$. The fourth is $4^3 - 1 = 64 - 1 = 63$. The fifth is $5^3 - 1 = 125 - 1 = 124$. The next term will be the sixth term: $6^3 - 1 = 216 - 1 = 215$.

Question 10

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: $4 \times 2 + 3 = 8 + 3 = 11$. $11 \times 2 + 3 = 22 + 3 = 25$. $25 \times 2 + 3 = 50 + 3 = 53$. To find the next term, we apply the rule to 53: $53 \times 2 + 3 = 106 + 3 = 109$.

Question 11

1. 35
2. 36
3. 37 (correct answer)
4. 38

Explanation: The pattern for this sequence is $n^2 + 1$, where n is the term's position in the sequence (starting with n=1). The first term is $1^2 + 1 = 2$. The second term is $2^2 + 1 = 5$. The third term is $3^2 + 1 = 10$. The fourth term is $4^2 + 1 = 17$. The fifth term is $5^2 + 1 = 26$. The next term (the sixth term) is $6^2 + 1 = 36 + 1 = 37$.

Question 12

1. 77
2. 76
3. 75 (correct answer)
4. 73

Explanation: The differences between consecutive terms are increasing. The differences are: $100 - 99 = 1$, $99 - 96 = 3$, $96 - 91 = 5$, $91 - 84 = 7$. The amounts being subtracted are consecutive odd numbers. The next odd number to subtract is 9. Therefore, the next term is $84 - 9 = 75$.

Question 13

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. $10 + 10 = 20$; $20 - 5 = 15$; $15 + 10 = 25$; $25 - 5 = 20$; $20 + 10 = 30$. The next step is to subtract 5: $30 - 5 = 25$.

Question 14

1. $\frac{8}{14}$
2. $\frac{9}{13}$
3. $\frac{9}{14}$ (correct answer)
4. $\frac{11}{14}$

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 $11 + 3 = 14$. Therefore, the next fraction in the sequence is $\frac{9}{14}$.

Question 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

1. 37
2. 41 (correct answer)
3. 45
4. 36

Explanation: The sequence is an arithmetic sequence with a first term $a_1 = 5$. The common difference $d$ is $9 - 5 = 4$. The formula for the nth term of an arithmetic sequence is $a_n = a_1 + (n-1)d$. To find the 10th term ($a_{10}$), we substitute the values: $a_{10} = 5 + (10-1) \times 4 = 5 + 9 \times 4 = 5 + 36 = 41$.

Question 17

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: $18 \div 12 = 1.5$. Let's check with another pair: $27 \div 18 = 1.5$. The common ratio is 1.5. To find the next term, multiply the last term by 1.5: $40.5 \times 1.5 = 60.75$.

Question 18

1. 51
2. 55
3. 60 (correct answer)
4. 61

Explanation: Let's examine the differences between consecutive terms: $6-5=1$, $10-6=4$, $19-10=9$, $35-19=16$. The differences are 1, 4, 9, 16, which are the perfect squares $1^2, 2^2, 3^2, 4^2$. The next difference in the pattern will be $5^2 = 25$. Adding this to the last term gives the next term in the sequence: $35 + 25 = 60$.

Question 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. $2 + 5 = 7$, $5 + 7 = 12$, $7 + 12 = 19$. The next term is found by adding the last two terms: $12 + 19 = 31$.

Question 20

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. $400 \div 4 = 100$; $100 \div 4 = 25$; $25 \div 4 = 6.25$. To find the next term, we divide 6.25 by 4: $6.25 \div 4 = 1.5625$.