Numerical Sequence Patterns - SSAT Upper Level Quantitative

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

   $5$. The common difference is calculated by subtracting consecutive terms, yielding a constant $5$.

2. What is the next term in the geometric sequence $3, 6, 12, 24, \dots$?

   $48$. The sequence is geometric with a common ratio of $2$, so the next term is obtained by multiplying $24$ by $2$.

3. What is the next term in the sequence $10, 7, 14, 11, 22, 19, \dots$?

   $38$. The pattern alternates subtracting $3$ and multiplying by $2$, applying multiplication by $2$ to $19$.

4. What is the common ratio in the geometric sequence $81, 27, 9, 3, \dots$?

   $\frac{1}{3}$. The common ratio is determined by dividing consecutive terms, resulting in $\frac{1}{3}$ consistently.

5. What is the next term in the sequence $3, 7, 15, 31, 63, \dots$?

   $127$. Each term is one less than a power of $2$, specifically $2^{n+1} - 1$, with the next for $n=6$.

6. What is the next term in the sequence of perfect squares $1, 4, 9, 16, 25, \dots$?

   $36$. The sequence consists of squares of consecutive integers, with the next being $6^2$.

7. What is the next term in the sequence $-1, 2, -4, 8, -16, \dots$?

   $32$. The geometric sequence has a common ratio of $-2$, alternating signs while doubling in magnitude.

8. What is the next term in the sequence $100, 50, 25, 12.5, \dots$?

   $6.25$. The geometric sequence has a common ratio of $\frac{1}{2}$, halving each term.

9. What is the next term in the sequence $1, 3, 6, 10, 15, 21, \dots$?

   $28$. This is the sequence of triangular numbers, where each adds the next integer, so $21 + 7$.

10. What is the next term in the sequence of triangular numbers $1, 3, 6, 10, 15, \dots$?

    $21$. Triangular numbers sum the first $n$ natural numbers, so the next adds $6$ to $15$.

11. What is the next term in the sequence with increasing differences $1, 2, 4, 7, 11, \dots$?

    $16$. Differences increase by $1$ each time starting from $1$, so add $5$ to $11$.

12. What is the next term in the sequence $4, 9, 16, 25, 36, \dots$?

    $49$. The sequence is squares of integers starting from $2$, with the next being $7^2$.

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

    $21$. Each term is the sum of the two preceding ones, following a Fibonacci-like pattern starting from $2$ and $3$.

14. What is the next term in the sequence $2, 5, 10, 17, 26, \dots$?

    $37$. Differences are odd numbers starting from $3$ and increasing by $2$, so add $11$ to $26$.

15. What is the next term in the arithmetic sequence $7, 12, 17, 22, \dots$?

    $27$. The sequence is arithmetic with a common difference of $5$, so the next term is found by adding $5$ to $22$.

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

    $a_n = a_1 r^{n-1}$. This formula computes the $n$th term by multiplying the first term $a_1$ by the common ratio $r$ raised to the power of $n-1$.

17. What is the next term in the alternating sequence $5, -5, 5, -5, \dots$?

    $5$. The sequence alternates between positive and negative $5$, so after $-5$ it returns to positive.

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

    $33$. Each term is twice the previous minus $1$, applying the rule to $17$.

19. What is the next term in the sequence of perfect cubes $1, 8, 27, 64, \dots$?

    $125$. The sequence comprises cubes of consecutive integers, with the next being $5^3$.

20. What is the next term in the sequence $2, 4, 7, 11, 16, \dots$?

    $22$. Differences increase by $1$ starting from $2$, so add $6$ to $16$.

21. What is the explicit formula for the $n$th triangular number $T_n$?

    $T_n = \frac{n(n+1)}{2}$. The formula sums the first $n$ positive integers to yield the $n$th triangular number.

22. What is the $n$th-term formula for an arithmetic sequence with first term $a_1$ and common difference $d$?

    $a_n = a_1 + (n-1)d$. This formula derives the $n$th term by starting from $a_1$ and adding the common difference $d$ for each subsequent term up to $n-1$.

23. What is the next term in the sequence $1, 4, 2, 8, 4, 16, \dots$?

    $8$. The pattern alternates between multiplying by $4$ and dividing by $2$, so divide $16$ by $2$.

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