Missing Terms in Sequences - ISEE Upper Level Quantitative Reasoning

1. What is the missing term in the arithmetic sequence $-3, 1, 5, \_, 13$?

   $9$. The common difference in this arithmetic sequence is 4, so add 4 to 5 to find the missing term.

2. What is the missing term in the arithmetic sequence $12, \_, 4, 0, -4$?

   $8$. The common difference in this arithmetic sequence is -4, so subtract 4 from 12 to find the missing term.

3. What is the missing term in the geometric sequence $\frac{3}{2}, \frac{3}{4}, \_, \frac{3}{16}$?

   $\frac{3}{8}$. The common ratio in this geometric sequence is $\frac{1}{2}$, so multiply $\frac{3}{4}$ by $\frac{1}{2}$ to find the missing term.

4. What is the missing term in the Fibonacci-type sequence $1, 1, 2, 3, 5, \_, 13$?

   $8$. In this Fibonacci sequence, each term is the sum of the two preceding ones, so add 3 and 5.

5. What is the missing term in the arithmetic sequence $\frac{1}{2}, \frac{5}{2}, \frac{9}{2}, \_, \frac{17}{2}$?

   $\frac{13}{2}$. The common difference in this arithmetic sequence is 2, so add 2 to $\frac{9}{2}$ to find the missing term.

6. What is the missing term in the square-number sequence $1, 4, 9, \_, 25$?

   $16$. This sequence consists of squares of consecutive integers, so the fourth term is $4^2$.

7. What is the missing term in the sequence $3, 7, 13, 21, \_, 43$ where each term is $n^2+n+1$?

   $31$. Each term follows the pattern $n^2 + n + 1$ for consecutive integers n starting from 1, so use n=5.

8. What is the missing term in the arithmetic sequence $5, 9, 13, \_, 21$?

   $17$. The common difference in this arithmetic sequence is 4, so add 4 to 13 to find the missing term.

9. What is the missing term in the sequence $2, 3, 5, 8, 12, \_, 23$ where differences increase by $1$?

   $17$. Differences increase by 1 starting from 1, so add 5 to 12 to find the missing term.

10. What is the missing term in the sequence $100, 50, 25, \_, \frac{25}{4}$ where each term is halved?

    $\frac{25}{2}$. Each term is half the previous in this geometric sequence with ratio $\frac{1}{2}$, so divide 25 by 2.

11. What is the missing term in the sequence $2, 4, 8, 16, \_, 64$ where each term doubles?

    $32$. The common ratio in this geometric sequence is 2, so multiply 16 by 2 to find the missing term.

12. What is the missing term in the geometric sequence $81, 27, 9, \_, 1$?

    $3$. The common ratio in this geometric sequence is $\frac{1}{3}$, so multiply 9 by $\frac{1}{3}$ to find the missing term.

13. What is the missing term in the sequence $1, 2, 4, 7, 11, \_, 22$ where differences are $1,2,3,4,\dots$?

    $16$. Differences increase by 1 starting from 1, so add 5 to 11 to find the missing term.

14. What is the missing term in the cube-number sequence $1, 8, \_, 64, 125$?

    $27$. This sequence consists of cubes of consecutive integers, so the third term is $3^3$.

15. What is the missing term in the geometric sequence $2, 6, 18, \_, 162$?

    $54$. The common ratio in this geometric sequence is 3, so multiply 18 by 3 to find the missing term.

16. What is the missing term in the triangular-number sequence $1, 3, 6, 10, \_, 21$?

    $15$. Triangular numbers are sums of the first n integers, so the fifth is $\frac{5 \times 6}{2}$.

17. What is the missing term in the sequence $9, 7, 10, 8, 11, \_, 12$ that alternates $-2, +3$?

    $9$. The pattern alternates subtracting 2 and adding 3, so subtract 2 from 11 to find the missing term.

18. What is the missing term in the sequence $6, 11, 21, 41, \_, 161$ where each term is $2x-1$?

    $81$. Each term is twice the previous minus 1, so apply the rule to 41 to find the missing term.

19. What is the missing term in the sequence $2, 5, 10, 17, \_, 37$ where each term is $n^2+1$?

    $26$. Each term follows the pattern $n^2 + 1$ for consecutive integers n starting from 1, so use n=5.

20. What is the missing term in the sequence $1, 2, 6, 24, \_, 720$?

    $120$. This sequence follows the factorial pattern $n!$, so the fifth term is $5!$.

21. What is the missing term in the sequence $1, 3, 7, 15, \_, 63$ where each term is $2^n-1$?

    $31$. Each term follows the pattern $2^n - 1$ for consecutive integers n starting from 1, so use n=5.

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