Pattern Rules From Sequences - SSAT Middle Level Quantitative

1. What is the pattern rule for the sequence $1, 3, 6, 10, 15, \dots$?

   Add $2, 3, 4, 5, \dots$ (triangular numbers). These triangular numbers are formed by adding consecutive integers starting from $2$ after the initial $1$.

2. What is the pattern rule for the sequence $5, 10, 8, 16, 14, 28, \dots$?

   Alternate $\times 2$ and $-2$. Operations cycle between multiplying by $2$ and subtracting $2$, generating the sequence's progression.

3. What is the explicit rule for the geometric sequence with $a_1 = 5$ and common ratio $r = 2$?

   $a_n = 5 \cdot 2^{n - 1}$. This expresses the $n$th term of a geometric sequence using the initial term and common ratio.

4. What is the pattern rule for the sequence $-1, 1, -1, 1, -1, \dots$?

   Multiply by $-1$ each term. Alternating signs are produced by negating each previous term, creating an oscillating pattern.

5. What is the pattern rule for the sequence $9, 3, 1, \frac{1}{3}, \frac{1}{9}, \dots$?

   Multiply by $\frac{1}{3}$ each term. Each term is obtained by multiplying the prior one by $\frac{1}{3}$, forming a decreasing geometric sequence.

6. What is the pattern rule for the sequence $1, 2, 4, 7, 11, 16, \dots$?

   Add $1, 2, 3, 4, 5, \dots$. The sequence is generated by adding consecutive positive integers starting from $1$ to the previous term.

7. What is the pattern rule for the sequence $2, 6, 18, 54, \dots$?

   Multiply by $3$ each term. Tripling happens between terms, creating a geometric sequence with ratio $3$.

8. What is the pattern rule for the sequence $6, 11, 16, 21, 26, \dots$?

   Add $5$ each term. An arithmetic sequence with a constant difference of $5$ added to each preceding term.

9. What is the pattern rule for the sequence $2, 6, 12, 20, 30, \dots$?

   Add consecutive even numbers: $+4, +6, +8, +10, \dots$. Differences between consecutive terms are even numbers increasing by $2$ starting from $4$, building the sequence cumulatively.

10. What is the pattern rule for the sequence $10, 7, 4, 1, -2, \dots$?

    Subtract $3$ each term. An arithmetic sequence with a common difference of $-3$, resulting in a steady decrease.

11. What is the pattern rule for the sequence $81, 27, 9, 3, 1, \dots$?

    Divide by $3$ each term. This geometric sequence decreases as each term is one-third of the previous, achieved by repeated division by $3$.

12. What is the pattern rule for the sequence $1, 1, 2, 3, 5, 8, \dots$?

    Each term is the sum of the previous two terms. This is the Fibonacci sequence starting with two $1$s, where terms sum the previous two.

13. What is the pattern rule for the sequence $0, 1, 0, 1, 0, 1, \dots$?

    Alternate $0$ and $1$. The pattern oscillates between $0$ and $1$ in a repeating binary fashion for every pair of terms.

14. What is the pattern rule for the sequence $4, 7, 11, 16, 22, \dots$?

    Add $3, 4, 5, 6, \dots$ (increasing by $1$ each time). Differences start at $3$ and increase by $1$ each step, accumulating to form the sequence.

15. What is the pattern rule for the sequence $2, 1, 2, 1, 2, 1, \dots$?

    Alternate $2$ and $1$. The pattern repeatedly switches between $2$ and $1$ in an alternating manner for each term.

16. What is the pattern rule for the sequence $1, 3, 1, 3, 1, 3, \dots$?

    Alternate $+2$ and $-2$. The sequence switches between $1$ and $3$ by alternately adding and subtracting $2$ from the previous term.

17. What is the pattern rule for the sequence $1, 8, 27, 64, 125, \dots$?

    Cubes: $n^3$ for $n = 1, 2, 3, \dots$. Terms are cubes of consecutive positive integers, starting from $1^3$ and increasing accordingly.

18. What is the explicit rule for the arithmetic sequence with $a_1 = 7$ and common difference $d = 3$?

    $a_n = 7 + 3(n - 1)$. The formula calculates the $n$th term of an arithmetic sequence using the first term and common difference.

19. What is the pattern rule for the sequence $3, 6, 12, 24, 48, \dots$?

    Multiply by $2$ each term. Doubling occurs between terms, establishing a geometric sequence with a constant ratio of $2$.

20. What is the pattern rule for the sequence $5, 9, 13, 17, \dots$?

    Add $4$ each term. The sequence forms an arithmetic progression with a common difference of $4$, increasing steadily by this constant each time.

21. What is the pattern rule for the sequence $2, 4, 8, 16, 32, \dots$?

    Powers of $2$: $2^1, 2^2, 2^3, \dots$. Terms represent powers of $2$ where the exponent matches the term's position in the sequence.

22. What is the pattern rule for the sequence $2, 5, 10, 17, 26, \dots$?

    Add consecutive odd numbers: $+3, +5, +7, +9, \dots$. Consecutive odd integers beginning from $3$ are added to build each subsequent term.

23. What is the pattern rule for the sequence $2, 3, 5, 8, 13, 21, \dots$?

    Each term is the sum of the previous two terms. This resembles the Fibonacci sequence but starts at $2$ and $3$, with each subsequent term summing the prior two.

24. What is the pattern rule for the sequence $1, 4, 9, 16, 25, \dots$?

    Squares: $n^2$ for $n = 1, 2, 3, \dots$. Each term corresponds to the square of its position index, forming the sequence of perfect squares.

25. What is the pattern rule for the sequence $100, 50, 25, 12.5, \dots$?

    Multiply by $\frac{1}{2}$ each term. The sequence halves each time, creating a geometric progression with a common ratio of $\frac{1}{2}$.

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