Number Pattern Rules - ISEE Middle Level Mathematics Achievement

1. What rule generates the pattern $2, 5, 10, 17, 26, \dots$?

   Add consecutive odd numbers: $+3,+5,+7,+9,\dots$. The differences between terms are consecutive odd numbers starting from $3$, building the sequence cumulatively.

2. What explicit rule gives the $n$th term of $0, 1, 3, 6, 10, \dots$?

   $a_n = \frac{n(n-1)}{2}$. This formula computes the $(n-1)$th triangular number, summing integers up to $n-1$.

3. What rule generates the pattern $1, 1, 2, 3, 5, 8, \dots$?

   Each term is the sum of the two previous terms. This follows the Fibonacci recurrence, where each term after the first two is the sum of the preceding pair.

4. What explicit rule gives the $n$th term of $2, 5, 10, 17, 26, \dots$?

   $a_n = n^2 + 1$. This formula adds $1$ to the square of $n$ to produce each term in the sequence.

5. What rule generates the pattern $1, 4, 9, 16, 25, \dots$?

   Square the term number: $a_n=n^2$. The pattern consists of perfect squares, where each term is the square of its position $n$.

6. What rule generates the pattern $4, 9, 16, 25, 36, \dots$?

   Square the term number plus $1$: $a_n=(n+1)^2$. Each term is the square of $(n+1)$, producing shifted perfect squares starting from $4$.

7. What explicit rule gives the $n$th term of $64, 32, 16, 8, \dots$?

   $a_n = 64\cdot \left(\frac{1}{2}\right)^{n-1}$. This formula expresses the geometric sequence starting at $64$ with common ratio $\frac{1}{2}$ for the $n$th term.

8. What explicit rule gives the $n$th term of $3, 7, 11, 15, \dots$?

   $a_n = 3 + 4(n-1)$. This formula captures the arithmetic sequence starting at $3$ with a common difference of $4$ to find the $n$th term directly.

9. What rule generates the pattern $1, 2, 4, 7, 11, \dots$?

   Add $1,2,3,4,\dots$. The differences are consecutive positive integers starting from $1$, accumulating to form the sequence.

10. What recursive rule defines $1, 1, 2, 3, 5, 8, \dots$?

    $a_1=1$, $a_2=1$, and $a_n=a_{n-1}+a_{n-2}$. This defines the Fibonacci sequence starting with two $1$s and using the sum of the previous two terms for each subsequent one.

11. What explicit rule gives the $n$th term of $5, 10, 20, 40, \dots$?

    $a_n = 5\cdot 2^{n-1}$. This formula defines the geometric sequence with first term $5$ and common ratio $2$ for the $n$th term.

12. What rule generates the pattern $2, 6, 12, 20, 30, \dots$?

    Products of consecutive integers: $a_n=n(n+1)$. Each term is the product of $n$ and $(n+1)$, generating twice the triangular numbers.

13. What explicit rule gives the $n$th term of $3, -3, 3, -3, \dots$?

    $a_n = 3(-1)^{n-1}$. The formula uses powers of $-1$ to alternate signs, starting positive for odd $n$ and negative for even $n$.

14. What explicit rule gives the $n$th term of $7, 12, 17, 22, \dots$?

    $a_n = 7 + 5(n-1)$. This arithmetic formula uses the first term $7$ and common difference $5$ to find the $n$th term.

15. What explicit rule gives the $n$th term of $20, 16, 12, 8, \dots$?

    $a_n = 20 - 4(n-1)$. This formula represents the arithmetic sequence starting at $20$ with a common difference of $-4$ for the $n$th term.

16. What recursive rule generates $2, 6, 18, 54, \dots$?

    $a_1=2$, and $a_n=3a_{n-1}$. This recursive definition starts with $2$ and multiplies each subsequent term by $3$, forming a geometric sequence.

17. What rule generates the arithmetic pattern $5, 9, 13, 17, \dots$?

    Add $4$ each term. The pattern is an arithmetic sequence with a common difference of $4$, increasing each term by that constant.

18. What rule generates the alternating pattern $2, 5, 2, 5, 2, \dots$?

    Repeat the cycle $2,5$. The pattern is periodic with period $2$, alternating between $2$ and $5$ indefinitely.

19. What rule generates the arithmetic pattern $18, 15, 12, 9, \dots$?

    Subtract $3$ each term. The pattern is an arithmetic sequence with a common difference of $-3$, decreasing each term by $3$.

20. What explicit rule gives the $n$th term of $1, 2, 4, 7, 11, \dots$?

    $a_n = 1 + \frac{n(n-1)}{2}$. This formula adds $1$ to the $(n-1)$th triangular number to generate each term.

21. What rule generates the pattern $1, 3, 6, 10, 15, \dots$?

    Add $2,3,4,5,\dots$. The differences start from $2$ and increase by $1$ each time, forming triangular numbers beginning at $1$.

22. What rule generates the pattern $0, 1, 3, 6, 10, \dots$?

    Add $1,2,3,4,\dots$ (triangular numbers). The pattern forms triangular numbers by cumulatively adding consecutive positive integers starting from $1$.

23. What rule generates the geometric pattern $81, 27, 9, 3, \dots$?

    Multiply by $\frac{1}{3}$ each term. The pattern is a geometric sequence with a common ratio of $\frac{1}{3}$, reducing each term accordingly.

24. What rule generates the pattern $1, 8, 27, 64, \dots$?

    Cube the term number: $a_n=n^3$. The pattern consists of perfect cubes, where each term is the cube of its position $n$.

25. What rule generates the alternating pattern $3, -3, 3, -3, \dots$?

    Multiply by $-1$ each term. Multiplying by $-1$ alternates the sign of each term while keeping the absolute value constant at $3$.

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