Missing Terms in Sequences - ISEE Middle Level Mathematics Achievement

What is the next term in the pattern: 100, 99, 96, 91, 84, ...?
Answer: 75 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 $84 - 9 = 75$ .
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?
Answer: 44 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.
Based on the given pattern, what number should replace the blank: 50, 40, $\underline{\hspace{2em}}$ , 20?
Answer: 30 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 50, 40, __, 20, the pattern involves subtracting 10 from each term (50-10=40, and we need to verify with 20). The correct answer is C (30) because it logically follows the pattern: 50-10=40, 40-10=30, 30-10=20. Choice A (25) is incorrect because it assumes subtracting 15, while choice D (35) assumes subtracting only 5.
A sequence follows an arithmetic pattern: -21, -15, $\underline{\hspace{2em}}$ , -3, 3. What is the missing term?
Answer: -9 In an arithmetic sequence, the difference between consecutive terms is constant. The difference is $-15 - (-21) = -15 + 21 = 6$ . Let's check with the last two terms: $3 - (-3) = 3 + 3 = 6$ . The common difference is 6. To find the missing term, add 6 to the term before it: $-15 + 6 = -9$ . We can check this: $-9 + 6 = -3$ , which is correct.
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, ...?
Answer: 0 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: $4 - 4 = 0$ .
A sequence has two missing terms: 2.3, 3.4, $\underline{\hspace{2em}}$ , $\underline{\hspace{2em}}$ , 6.7. If the sequence is arithmetic, what is the first of the two missing numbers?
Answer: 4.5 To find the common difference in an arithmetic sequence, we can find the total difference between two known terms and divide by the number of steps between them. The difference between 6.7 and 3.4 is $6.7 - 3.4 = 3.3$ . There are three steps between the term 3.4 and the term 6.7 (from 3.4 to the first blank, from the first blank to the second, and from the second blank to 6.7). So, the common difference is $3.3 \div 3 = 1.1$ . The first missing term is $3.4 + 1.1 = 4.5$ . The second missing term would be $4.5 + 1.1 = 5.6$ , which fits the pattern.
What is the next number in the following pattern? 5, 3, 9, 7, 21, 19, ...
Answer: 57 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: $19 \times 3 = 57$ .
Find the next term in the sequence: 3, 7, 14, 24, 37, ...
Answer: 53 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 $37+16=53$ .
Find the missing term in the sequence: 4, 1, 8, 4, 16, 7, 32, $\underline{\hspace{2em}}$ .
Answer: 10 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 $7 + 3 = 10$ .
Which number completes the sequence: 5, 9, $\underline{\hspace{2em}}$ , 17?
Answer: 13 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.
Find the next term in the sequence: 400, 100, 25, 6.25, ...
Answer: 1.5625 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: $6.25 \div 4 = 1.5625$ .
The first three terms of an arithmetic sequence are 5, 9, and 13. What is the 10th term in the sequence?
Answer: 41 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: $a_{10} = 5 + (10-1) \times 4 = 5 + 9 \times 4 = 5 + 36 = 41$ .
What is the missing term in the sequence: 30, 25, $\underline{\hspace{2em}}$ , 15?
Answer: 20 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.
Find the number that completes the sequence: 0, 7, 26, 63, 124, $\underline{\hspace{2em}}$ .
Answer: 215 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: $6^3 - 1 = 216 - 1 = 215$ .
What is the next number in the sequence? 2, 5, 7, 12, 19, ...
Answer: 31 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: $12 + 19 = 31$ .
What is the next term in the sequence 4, 11, 25, 53, ...?
Answer: 109 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: $53 \times 2 + 3 = 106 + 3 = 109$ .
Find the next number in the sequence: 12, 18, 27, 40.5, ...
Answer: 60.75 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: $40.5 \times 1.5 = 60.75$ .
Identify the missing term in the pattern: 8, 11, 14, $\underline{\hspace{2em}}$ .
Answer: 17 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.
Identify the missing term in the pattern: 9, 12, $\underline{\hspace{2em}}$ , 18.
Answer: 15 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 9, 12, __, 18, the pattern involves adding 3 to each term (12-9=3, and we need to verify 18-15=3). The correct answer is C (15) because it logically follows the pattern: 9+3=12, 12+3=15, 15+3=18. Choice A (13) is incorrect because it assumes adding only 1, while choice D (16) assumes adding 4.
If the pattern continues, what is the next fraction in the sequence? $\frac{1}{2}, \frac{3}{5}, \frac{5}{8}, \frac{7}{11}, ...$
Answer: $\frac{9}{14}$ 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 $\frac{9}{14}$ .

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