Pattern Rules - ISEE Middle Level Quantitative Reasoning

1. What is the rule for the sequence $\frac{1}{2}, 1, 2, 4, \dots$ in terms of $n$?

   $a_n = \frac{1}{2}\cdot 2^{n-1}$. The geometric sequence begins with $\frac{1}{2}$ and has a common ratio of 2, forming the explicit rule.

2. Identify the next term in the sequence $81, 27, 9, 3, 1, \dots$.

   $\frac{1}{3}$. The geometric sequence has a common ratio of $\frac{1}{3}$, so multiply the last term by $\frac{1}{3}$.

3. What is the rule for the sequence $2, 6, 18, 54, \dots$ in terms of $n$?

   $a_n = 2\cdot 3^{n-1}$. The geometric sequence starts with 2 and has a common ratio of 3, forming the rule.

4. Identify the next term in the sequence $12, 11, 9, 6, 2, \dots$.

   $-3$. The differences are -1, -2, -3, -4, decreasing by 1 each time, so subtract 5 from 2.

5. Identify the next term in the sequence $2, 5, 10, 17, 26, \dots$.

   $37$. The differences between terms are 3, 5, 7, 9, increasing by 2 each time, so the next difference is 11 added to 26.

6. Identify the next term in the sequence $5, 15, 45, 135, \dots$.

   $405$. The geometric sequence has a common ratio of 3, so multiply 135 by 3.

7. What is the rule for the sequence $0, 1, 4, 9, 16, \dots$ using $n$?

   $a_n = (n-1)^2$. Each term is the square of one less than its position number.

8. What is the rule for the sequence $10, 7, 4, 1, -2, \dots$ in terms of $n$?

   $a_n = 10 - 3(n-1)$. The sequence is arithmetic with first term 10 and common difference -3, yielding the explicit formula.

9. What is the common ratio for the sequence $160, 80, 40, 20, \dots$?

   $r = \frac{1}{2}$. Each term is half of the previous, defining the constant ratio.

10. What is the rule for the geometric sequence $3, 6, 12, 24, \dots$ in terms of $n$?

    $a_n = 3\cdot 2^{n-1}$. The sequence is geometric with first term 3 and common ratio 2, forming the explicit rule.

11. What is the pattern rule for $1, 4, 9, 16, 25, \dots$ using $n$?

    $a_n = n^2$. Each term is the square of its position number in the sequence.

12. What is the common difference for the sequence $-3, 2, 7, 12, \dots$?

    $d = 5$. Each term increases by 5 from the previous, establishing the constant difference.

13. Identify the next term in the sequence $4, 7, 13, 25, 49, \dots$.

    $97$. Each term is obtained by doubling the previous term and subtracting 1.

14. What is the explicit rule for the arithmetic sequence $-8, -3, 2, 7, \dots$?

    $a_n = -8 + 5(n-1)$. The arithmetic sequence has first term -8 and common difference 5, leading to the explicit rule.

15. What is the rule for the sequence $7, 14, 21, 28, \dots$ in terms of $n$?

    $a_n = 7n$. The sequence consists of multiples of 7, directly proportional to $n$.

16. Identify the next term in the sequence $1, 3, 6, 10, 15, \dots$.

    $21$. These are triangular numbers where each term is the sum of the first $n$ natural numbers, and the differences increase by 1, so add 6 to 15.

17. Identify the next term in the sequence $2, 4, 7, 11, 16, \dots$.

    $22$. The differences are 2, 3, 4, 5, increasing by 1 each time, so add 6 to 16.

18. Identify the next term in the sequence $2, 3, 5, 9, 17, \dots$.

    $33$. The differences are 1, 2, 4, 8, doubling each time, so add 16 to 17.

19. What is the rule for the sequence $1, 8, 27, 64, 125, \dots$ using $n$?

    $a_n = n^3$. Each term is the cube of its position number in the sequence.

20. What is the rule for the arithmetic sequence $5, 9, 13, 17, \dots$ in terms of $n$?

    $a_n = 5 + 4(n-1)$. The sequence is arithmetic with the first term 5 and common difference 4, so the explicit formula uses these values.

21. Identify the next term in the sequence $1, 1, 2, 3, 5, 8, \dots$.

    $13$. This is the Fibonacci sequence where each term is the sum of the two preceding ones.

22. What is the explicit rule for the sequence $2, 1, \frac{1}{2}, \frac{1}{4}, \dots$?

    $a_n = 2\cdot \left(\frac{1}{2}\right)^{n-1}$. The geometric sequence starts with 2 and has a common ratio of $\frac{1}{2}$, yielding the explicit formula.

23. Identify the next term in the sequence $3, 6, 10, 15, 21, \dots$.

    $28$. The differences are 3, 4, 5, 6, increasing by 1, so add 7 to 21.

24. What is the rule for the sequence $9, 6, 4, \frac{8}{3}, \dots$ in terms of $n$?

    $a_n = 9\cdot \left(\frac{2}{3}\right)^{n-1}$. The geometric sequence has first term 9 and common ratio $\frac{2}{3}$, forming the rule.

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