Arithmetic and Geometric Patterns - ISEE Upper Level Quantitative Reasoning

1. What is the common ratio $r$ for the geometric sequence $3, -6, 12, -24, \dots$?

   $r=-2$. Divide consecutive terms to determine the constant ratio of -2 in this geometric sequence.

2. What is the next term in the arithmetic sequence $18, 13, 8, 3, \dots$?

   $-2$. Subtract the common difference of -5 from the last term to obtain the next term.

3. What is the common difference $d$ for the arithmetic sequence $-4, 1, 6, 11, \dots$?

   $d=5$. Subtract consecutive terms to find the constant difference of 5 in this arithmetic sequence.

4. Which term equals $32$ in the geometric sequence $a_n=2\cdot 2^{n-1}$?

   $n=5$. Set the geometric formula equal to 32 and solve for $n$ to determine the term number.

5. What is the explicit formula for the $n$th term of a geometric sequence with first term $a_1$ and ratio $r$?

   $a_n=a_1\cdot r^{n-1}$. The formula calculates the $n$th term by multiplying the first term by the ratio raised to $(n-1)$.

6. What is the next term in the geometric sequence $160, 80, 40, 20, \dots$?

   $10$. Multiply the last term by the common ratio of $\frac{1}{2}$ to find the subsequent term.

7. What is the next term in the geometric sequence $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$?

   $\frac{1}{32}$. Multiply the last term by the common ratio of $\frac{1}{2}$ in this geometric sequence.

8. What is the next term in the Fibonacci-type pattern $2, 3, 5, 8, 13, \dots$?

   $21$. Add the last two terms, 8 and 13, to generate the next in this Fibonacci-like sequence.

9. Identify whether $0.5, 1, 2, 4, \dots$ is arithmetic or geometric.

   Geometric. A constant ratio of 2 between terms confirms the sequence is geometric.

10. Identify the pattern type for $1, 1, 2, 3, 5, 8, \dots$ (each term from previous terms).

    Fibonacci-type: $a_n=a_{n-1}+a_{n-2}$. Each term is the sum of the two preceding terms, defining a Fibonacci sequence variant.

11. What is the next term in the pattern $2, 6, 12, 20, \dots$?

    $30$. Each term follows the pattern $n(n+1)$ for increasing values of $n$ starting from 1.

12. What is the next term in the geometric sequence $-81, 27, -9, 3, \dots$?

    $-1$. Multiply the last term by the common ratio of $-\frac{1}{3}$ to find the next term.

13. What is the common difference $d$ if an arithmetic sequence has consecutive terms $x$ and $y$?

    $d=y-x$. Subtract the first term from the second to find the constant difference in an arithmetic sequence.

14. What is the next term in the pattern $1, 4, 9, 16, \dots$?

    $25$. The pattern consists of squares of consecutive integers: $1^2, 2^2, 3^2, 4^2, 5^2$.

15. Find the missing term in the geometric sequence $4, \_, 36$.

    $12$. The middle term is the geometric mean of 4 and 36, since ratios are constant.

16. Identify whether $5, 15, 45, 135, \dots$ is arithmetic or geometric.

    Geometric. The constant ratio of 3 between terms identifies the sequence as geometric.

17. What is the next term in the alternating pattern $3, -1, 3, -1, \dots$?

    $3$. The sequence alternates between 3 and -1, so it repeats 3 after -1.

18. Which term equals $0$ in the arithmetic sequence $a_n=12-3(n-1)$?

    $n=5$. Set the arithmetic formula equal to 0 and solve for $n$ to identify the term position.

19. Find the missing term in the arithmetic sequence $9, \_, 17$.

    $13$. The middle term is the arithmetic mean of 9 and 17, as the differences are equal.

20. What is $a_7$ for the geometric sequence with $a_1=2$ and $r=3$?

    $a_7=1458$. Apply the geometric sequence formula with $a_1=2$, $r=3$, and $n=7$ to find the term.

21. Identify whether $7, 12, 17, 22, \dots$ is arithmetic or geometric.

    Arithmetic. The constant difference of 5 between terms classifies the sequence as arithmetic.

22. What is $a_{10}$ for the arithmetic sequence with $a_1=4$ and $d=3$?

    $a_{10}=31$. Substitute $a_1=4$, $d=3$, and $n=10$ into the arithmetic sequence formula to compute the term.

23. What is the common ratio $r$ if a geometric sequence has consecutive terms $x$ and $y$?

    $r=\frac{y}{x}$. Divide the second term by the first to obtain the constant ratio in a geometric sequence.

24. What is the explicit formula for the $n$th term of an arithmetic sequence with first term $a_1$ and difference $d$?

    $a_n=a_1+(n-1)d$. The formula computes the $n$th term by adding $(n-1)$ times the common difference to the first term.

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