Number Pattern Rules - ISEE Lower Level Mathematics Achievement

1. What is the rule for $1, 3, 6, 10, 15, \dots$ (state $a_n$ in terms of $n$)?

   $a_n = \frac{n(n+1)}{2}$. Triangular numbers arise from the sum of first $n$ positives, given by this quadratic formula.

2. Identify the rule for $1, 1, 2, 3, 5, 8, \dots$ (state the recursive rule).

   $a_1=1,\ a_2=1,\ a_n=a_{n-1}+a_{n-2}$. The Fibonacci sequence is defined recursively by summing the two preceding terms after initial values.

3. Identify the rule for $64, 32, 16, 8, 4, \dots$ (state $a_n$ in terms of $n$).

   $a_n = 64\left(\frac{1}{2}\right)^{n-1}$. Geometric sequence starting at 64 with ratio $\frac{1}{2}$, following the explicit geometric form.

4. Which option is the common difference for the arithmetic sequence $-4, -1, 2, 5, \dots$?

   $3$. The constant difference between consecutive terms in this arithmetic sequence is 3.

5. What is the next term if the rule is $a_n = 7 + 2(n-1)$ and $a_4$ is the last shown term?

   $a_5 = 15$. Applying the arithmetic rule for $n=5$ gives the subsequent term after $a_4$.

6. Identify the rule for $5, 10, 20, 40, 80, \dots$ (state $a_n$ in terms of $n$).

   $a_n = 5\cdot 2^{n-1}$. Geometric sequence with initial 5 and ratio 2, using the standard explicit formula.

7. What is the rule for $1, 3, 5, 7, 9, \dots$ (state $a_n$ in terms of $n$)?

   $a_n = 2n - 1$. This represents odd numbers, derived as twice the index minus 1.

8. What is the rule for the alternating sequence $-2, 2, -2, 2, \dots$ (state $a_n$)?

   $a_n = 2(-1)^n$. The pattern alternates signs with magnitude 2, achieved by scaling $(-1)^n$.

9. Identify the rule for $81, 27, 9, 3, 1, \dots$ (state $a_n$ in terms of $n$).

   $a_n = 3^{5-n}$. This geometric sequence starts at 81 with common ratio $\frac{1}{3}$, expressible as decreasing powers of 3.

10. What is the rule for the sequence $12, 9, 6, 3, 0, \dots$ (state $a_n$ in terms of $n$)?

    $a_n = 15 - 3n$. This arithmetic sequence has first term 12 and common difference -3, derived from $a_n = a_1 + (n-1)d$.

11. Identify the rule for $2, 5, 10, 17, 26, \dots$ (state $a_n$ in terms of $n$).

    $a_n = n^2 + 1$. Each term adds 1 to the square of its position, creating a quadratic pattern.

12. What is the rule for the sequence $3, 7, 11, 15, \dots$ (state $a_n$ in terms of $n$)?

    $a_n = 4n - 1$. The sequence is arithmetic with first term 3 and common difference 4, yielding the general term through the formula $a_n = a_1 + (n-1)d$.

13. Identify the rule for $48, 16, \frac{16}{3}, \frac{16}{9}, \dots$ (give the recursive rule).

    $a_1=48,\ a_n=\frac{1}{3}a_{n-1}$. Geometric sequence multiplies each term by $\frac{1}{3}$ starting from 48.

14. What is the explicit formula for an arithmetic sequence with $a_1=9$ and common difference $d=-4$?

    $a_n = 9 - 4(n-1)$. The explicit form for an arithmetic sequence uses first term and common difference in $a_n = a_1 + (n-1)d$.

15. What is the rule for $2, 6, 12, 20, 30, \dots$ (state $a_n$ in terms of $n$)?

    $a_n = n(n+1)$. Each term is the product of consecutive integers, forming a quadratic sequence.

16. What is the explicit formula for a geometric sequence with $a_1=6$ and common ratio $r=\frac{1}{2}$?

    $a_n = 6\left(\frac{1}{2}\right)^{n-1}$. The explicit geometric formula incorporates first term and common ratio as $a_n = a_1 r^{n-1}$.

17. Identify the rule for $2, 4, 8, 16, 32, \dots$ (state $a_n$ in terms of $n$).

    $a_n = 2^n$. The sequence is geometric with first term 2 and common ratio 2, simplifying to powers of 2.

18. Identify the rule for $1, 2, 4, 7, 11, \dots$ (give the recursive rule).

    $a_1=1,\ a_n=a_{n-1}+(n-1)$. Each term adds the previous index value, creating increasing differences.

19. Which option is the common ratio for the geometric sequence $3, -6, 12, -24, \dots$?

    $-2$. The constant ratio in this geometric sequence, alternating signs, is -2.

20. Identify the rule for $0, 1, 4, 9, 16, \dots$ (state $a_n$ in terms of $n$).

    $a_n = (n-1)^2$. The sequence shifts perfect squares by starting at 0 for $n=1$.

21. What is the rule for $1, 4, 9, 16, 25, \dots$ (state $a_n$ in terms of $n$)?

    $a_n = n^2$. Each term is the square of its position index, forming the sequence of perfect squares.

22. What is the rule for $2, 4, 6, 8, 10, \dots$ (state $a_n$ in terms of $n$)?

    $a_n = 2n$. Even numbers are generated by multiplying the index by 2.

23. What is the rule for $1, 8, 27, 64, 125, \dots$ (state $a_n$ in terms of $n$)?

    $a_n = n^3$. Terms are cubes of consecutive integers starting from 1.

24. What is the rule for the alternating sequence $1, -1, 1, -1, \dots$ (state $a_n$)?

    $a_n = (-1)^{n-1}$. Alternating signs are produced by raising -1 to an exponent that toggles parity.

25. Identify the rule for $2, 5, 8, 11, 14, \dots$ (give the recursive rule).

    $a_1=2,\ a_n=a_{n-1}+3$. This arithmetic sequence adds a constant difference of 3 to each previous term starting from 2.

Study these flashcards