SSAT Upper Level Quiz: Numerical Sequence Patterns

5 questions · exam conditions

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Numerical Sequence PatternsQuestion 1 of 5

In a sequence, the first term is $3$ , and each subsequent term is obtained by multiplying the previous term by $2$ and then subtracting $1$ . What is the difference between the 5th term and the 4th term?

22

23

45

47

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SSAT Upper Level Quiz

SSAT Upper Level Quiz: Numerical Sequence Patterns

Practice Numerical Sequence Patterns in SSAT Upper Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Numerical Sequence Patterns, giving you a quick way to practice the rules, question types, and explanations that matter most for SSAT Upper Level.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$22$
$23$ (correct answer)
$45$
$47$

Explanation: The sequence follows the pattern:

a_n = 2a_{n-1} - 1

with

a_1 = 3

. Computing:

a_1 = 3

a_2 = 2(3) - 1 = 5

a_3 = 2(5) - 1 = 9

a_4 = 2(9) - 1 = 17

a_5 = 2(17) - 1 = 33

. The difference is

a_5 - a_4 = 33 - 17 = 23

. Choice A comes from incorrectly computing

a_4 = 16

and

a_5 = 31

, giving difference

22

. Choice C is

a_5 + 12 = 45

. Choice D is

a_4 + 30 = 47

Question 2

Consider the sequence $2, 6, 18, 54, 162, \ldots$ and the sequence $1, 5, 25, 125, 625, \ldots$ . If the $n$ th terms of these sequences are added together, what type of sequence is formed?

Arithmetic with common difference $4$
Geometric with common ratio $5$
Neither arithmetic nor geometric (correct answer)
Arithmetic with common difference $3$

Explanation: The first sequence is

2 \cdot 3^{n-1}

(geometric with ratio

3

). The second sequence is

5^{n-1}

(geometric with ratio

5

). The sum sequence is

2 \cdot 3^{n-1} + 5^{n-1}

3, 11, 43, 179, 787, \ldots

. Differences:

8, 32, 136, 608, \ldots

(not constant). Ratios:

\frac{11}{3} \approx 3.67

\frac{43}{11} \approx 3.91

\frac{179}{43} \approx 4.16

(not constant). The sequence is neither arithmetic nor geometric. Choices A and D assume it's arithmetic. Choice B assumes it's geometric.

Question 3

A sequence begins: $2, 7, 17, 32, 52, \ldots$ If this pattern continues, which expression represents the $n$ th term for $n \geq 3$ ?

$5n^2 - 13n + 10$
$4n^2 - 11n + 9$
$3n^2 - 8n + 7$
$2.5n^2 - 2.5n + 2$ (correct answer)

Explanation: When you encounter a sequence problem, you need to identify the pattern by examining the differences between consecutive terms. This sequence tests your ability to recognize when a quadratic expression generates the terms. Let's find the pattern by calculating the first and second differences: First differences:

7-2=5

17-7=10

32-17=15

52-32=20

Second differences:

10-5=5

15-10=5

20-15=5

Since the second differences are constant (5), this confirms the sequence follows a quadratic pattern of the form

an^2 + bn + c

. To verify the correct answer, test choice D with the given terms:

For $n=1$ : $2.5(1)^2 - 2.5(1) + 2 = 2.5 - 2.5 + 2 = 2$ ✓
For $n=2$ : $2.5(4) - 2.5(2) + 2 = 10 - 5 + 2 = 7$ ✓
For $n=3$ : $2.5(9) - 2.5(3) + 2 = 22.5 - 7.5 + 2 = 17$ ✓

Choice A gives incorrect values: for

n=1

5(1) - 13(1) + 10 = 2

, but for

n=2

5(4) - 13(2) + 10 = 16 \neq 7

. Choice B similarly fails: for

n=2

4(4) - 11(2) + 9 = 11 \neq 7

. Choice C also produces wrong values: for

n=2

3(4) - 8(2) + 7 = 7

, but for

n=3

3(9) - 8(3) + 7 = 10 \neq 17

. Strategy tip: When you see constant second differences, you know you're dealing with a quadratic sequence. Always test your answer choice with multiple terms from the sequence to confirm it works consistently.

Question 4

A sequence is defined by $a_1 = 2$ , $a_2 = 5$ , and $a_n = 3a_{n-1} - 2a_{n-2}$ for $n \geq 3$ . Which term first exceeds $100$ ?

The 6th term
The 7th term (correct answer)
The 8th term
The 9th term

Explanation: Computing the sequence:

a_1 = 2

a_2 = 5

a_3 = 3(5) - 2(2) = 11

a_4 = 3(11) - 2(5) = 23

a_5 = 3(23) - 2(11) = 47

a_6 = 3(47) - 2(23) = 95

a_7 = 3(95) - 2(47) = 191

. The 7th term is the first to exceed

100

. Choice A gives

a_6 = 95 < 100

. Choices C and D are later terms that also exceed

100

but are not the first.

Question 5

A geometric sequence has its 3rd term equal to $18$ and its 6th term equal to $486$ . If all terms are positive, what is the 4th term?

$54$ (correct answer)
$36$
$72$
$81$

Explanation: When you encounter geometric sequence problems, remember that each term is found by multiplying the previous term by a constant ratio

r

. The general formula is

a_n = a_1 \cdot r^{n-1}

, where

a_1

is the first term. Given that the 3rd term is

18

and the 6th term is

486

, you can write:

$a_3 = a_1 \cdot r^2 = 18$
$a_6 = a_1 \cdot r^5 = 486$

To find the common ratio, divide the second equation by the first:

\frac{a_6}{a_3} = \frac{a_1 \cdot r^5}{a_1 \cdot r^2} = r^3 = \frac{486}{18} = 27

Taking the cube root:

r = 3

Now you can find

a_1

18 = a_1 \cdot 3^2 = 9a_1

, so

a_1 = 2

Therefore,

a_4 = a_1 \cdot r^3 = 2 \cdot 3^3 = 2 \cdot 27 = 54

. The answer is A. Looking at the wrong choices: B)

36

would result from incorrectly calculating

r = 2

instead of

3

. C)

72

might come from miscalculating

a_1 = 8

instead of

2

. D)

81

would result from thinking

a_4 = r^4 = 3^4

, forgetting to include the first term coefficient. Strategy tip: In geometric sequence problems, always use the ratio between given terms to find

r

first. When terms aren't consecutive, remember that

\frac{a_m}{a_n} = r^{m-n}

. This approach avoids having to solve for

a_1

until the end.