HSPT Quantitative Quiz: Identify Number Sequences

20 questions · exam conditions

0:00

Identify Number SequencesQuestion 1 of 20

4, 5, 10, 11, 33, 34, ?

102

136

137

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HSPT Quantitative Quiz

HSPT Quantitative Quiz: Identify Number Sequences

Practice Identify Number Sequences in HSPT Quantitative 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 Identify Number Sequences, giving you a quick way to practice the rules, question types, and explanations that matter most for HSPT Quantitative.

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

4, 5, 10, 11, 33, 34, ?

68
102
136 (correct answer)
137

Explanation: The pattern alternates adding 1 and multiplying by an increasing number: 4+1=5, 5×2=10, 10+1=11, 11×3=33, 33+1=34, then 34×4=136. The tempting 68 comes from repeating ×2, but the multiplier grows from 2 to 3 to 4.

Question 2

3, 7, 5, 11, 9, 17, 15, ?

21
23
25 (correct answer)
27

Explanation: The pattern alternates: add 4, subtract 2, add 6, subtract 2, add 8, subtract 2, so next add 10. 15 + 10 = 25. A tempting wrong answer is 23 if you only continue adding 8, but the added amount increases by 2 each time.

Question 3

64, 32, 36, 18, 22, ?

11 (correct answer)
18
26
44

Explanation: The sequence alternates between dividing by 2 and adding 4: 64 ÷ 2 = 32, 32 + 4 = 36, 36 ÷ 2 = 18, 18 + 4 = 22, so 22 ÷ 2 = 11. A tempting wrong answer is 44 because it looks like adding 22, but the actual next step is to halve 22.

Question 4

1, 2, 6, 15, 31, ?

47
52
55
56 (correct answer)

Explanation: Look at the gaps between terms: 1 to 2 is +1, 2 to 6 is +4, 6 to 15 is +9, and 15 to 31 is +16. These gaps are consecutive perfect squares, so the next gap is 25. Adding 25 to 31 gives 56. A tempting wrong answer is 47, which repeats the last gap of 16 instead of continuing the square pattern.

Question 5

2, 3, 8, 27, 112, ?

448
565 (correct answer)
560
564

Explanation: Multiply each term by an increasing whole number, then add that same number: 21+1=3, 32+2=8, 83+3=27, 274+4=112. Therefore, 112*5+5=565. The tempting 560 comes from multiplying by 5 but forgetting to add the 5.

Question 6

In the arithmetic sequence $7, 12, 17, 22, 27, ...$ what is the 20th term?

102 (correct answer)
97
107
112

Explanation: When you encounter an arithmetic sequence, you're looking at a pattern where each term increases by the same constant amount (called the common difference). In this sequence, each term increases by 5:

12 - 7 = 5

17 - 12 = 5

, and so on. To find any term in an arithmetic sequence, use the formula:

a_n = a_1 + (n-1)d

, where

a_n

is the nth term,

a_1

is the first term,

n

is the position, and

d

is the common difference. For the 20th term:

a_{20} = 7 + (20-1)(5) = 7 + 19(5) = 7 + 95 = 102

Looking at the answer choices, A) 102 is correct. B) 97 represents a common error where students forget to add the first term at the end—they calculate

19 \times 5 = 95

but mistakenly add 2 instead of 7, or simply miscalculate the final addition. C) 107 occurs when students use

n

instead of

(n-1)

in the formula, calculating

7 + 20(5) = 107

. This is a frequent mistake because students forget that the common difference is applied

(n-1)

times, not

n

times. D) 112 results from using the wrong formula entirely, perhaps calculating

7 + 21(5) = 112

, which adds an extra step to the common difference multiplication. Remember: In arithmetic sequences, always use

(n-1)

times the common difference, not

n

times. The first term already "uses up" one position, so you only need

(n-1)

additional steps of the common difference.

Question 7

The sequence $2, 6, 18, 54, 162, ...$ follows a pattern. If this pattern continues, what is the 8th term?

1458
1944
4374 (correct answer)
6561

Explanation: When you encounter a sequence like this, you're dealing with a geometric sequence where each term is found by multiplying the previous term by a constant ratio. Your first step is always to find this common ratio by dividing any term by the one before it. Let's find the pattern:

\frac{6}{2} = 3

\frac{18}{6} = 3

\frac{54}{18} = 3

. The common ratio is 3, meaning each term equals the previous term times 3. To find the 8th term, you can either multiply step by step or use the geometric sequence formula:

a_n = a_1 \times r^{(n-1)}

, where

a_1

is the first term,

r

is the common ratio, and

n

is the term number. Using the formula:

a_8 = 2 \times 3^7 = 2 \times 2187 = 4374

Or continuing the sequence: 2, 6, 18, 54, 162, 486, 1458, 4374. Looking at the wrong answers: Choice A (1458) is actually the 7th term—a classic off-by-one error that catches students who miscount. Choice B (1944) doesn't follow any clear pattern from this sequence. Choice D (6561) equals

3^8

, which represents what you'd get if you forgot to multiply by the initial term of 2. The correct answer is C) 4374. Study tip: For geometric sequences on the HSPT, always verify your common ratio with at least two pairs of consecutive terms, and double-check your term counting by writing out a few terms when in doubt.

Question 8

In the sequence $100, 94, 82, 64, 40, ...$ what is the pattern, and what comes next?

10 (correct answer)
8
12
6

Explanation: When you encounter a sequence problem, your first step is to look for patterns in how the numbers change from one term to the next. Don't assume it's a simple arithmetic sequence—sometimes the differences themselves follow a pattern. Let's examine the differences between consecutive terms in this sequence:

$100 - 94 = 6$
$94 - 82 = 12$
$82 - 64 = 18$
$64 - 40 = 24$

The differences are

6, 12, 18, 24

—each difference increases by 6! This means the next difference should be

24 + 6 = 30

. Therefore, the next term is

40 - 30 = 10

. Looking at the wrong answers: Choice B (8) might tempt you if you noticed the sequence decreases rapidly and guessed a small number, but it doesn't follow the established pattern. Choice C (12) could result from incorrectly thinking the differences repeat rather than increase systematically. Choice D (6) might come from using the first difference (6) without recognizing that the differences themselves are changing. The correct answer is A (10) because it follows the pattern where differences increase by 6 each time. Study tip: For HSPT sequence problems, always calculate the first few differences between terms. If those differences don't form an obvious pattern, check if the differences between the differences reveal a pattern. Many sequence problems involve this "second-order" pattern recognition rather than simple arithmetic progressions.

Question 9

In the sequence $3, 7, 15, 31, 63, \ldots$ , what is the $8$ th term?

$255$
$511$ (correct answer)
$127$
$383$

Explanation: When you encounter a sequence problem, your first step is identifying the pattern that connects consecutive terms. Look at how each term relates to the previous one. Let's examine the differences between consecutive terms:

7-3=4

15-7=8

31-15=16

63-31=32

. The differences are

4, 8, 16, 32

— each difference doubles! This means we add

4 \times 2^{n-1}

to get from the

n

th term to the

(n+1)

th term. Alternatively, you can spot another pattern: each term equals

2^{n+1} - 1

where

n

is the position. Check:

2^2-1=3

2^3-1=7

2^4-1=15

2^5-1=31

2^6-1=63

. Perfect! Using this formula, the 8th term is

2^{8+1}-1 = 2^9-1 = 512-1 = 511

. Choice (A)

255

equals

2^8-1

, which would be the 7th term using our formula — this catches students who miscount positions. Choice (C)

127

equals

2^7-1

, the actual 6th term, representing an even greater position error. Choice (D)

383

doesn't follow any clear pattern and likely represents a calculation mistake. The correct answer is (B)

511

. Strategy tip: For sequence problems, always look for patterns in differences between terms first. If the differences form a geometric sequence (like doubling), the original sequence often has a formula involving powers of 2. Double-check by verifying your pattern works for at least three given terms.

Question 10

What is the $12$ th term in the sequence: $1, 1, 2, 3, 5, 8, 13, 21, \ldots$ where each term after the second is the sum of the two preceding terms?

$89$
$144$ (correct answer)
$233$
$377$

Explanation: When you encounter a sequence where each term equals the sum of the two preceding terms, you're working with the famous Fibonacci sequence. This pattern appears frequently on standardized tests, so recognizing it immediately will save you time. To find the 12th term, continue the given pattern systematically. You have: 1, 1, 2, 3, 5, 8, 13, 21... Now add consecutive pairs:

9th term: $13 + 21 = 34$
10th term: $21 + 34 = 55$
11th term: $34 + 55 = 89$
12th term: $55 + 89 = 144$

The 12th term is

144

, which is choice B. Looking at the wrong answers: Choice A (

89

) is actually the 11th term—a common error when students miscount positions or stop one step too early. Choice C (

233

) is the 13th term, representing an off-by-one counting error in the opposite direction. Choice D (

377

) is the 14th term, showing a more significant counting mistake. These wrong answers aren't random; they're strategically placed Fibonacci numbers from nearby positions to catch counting errors. This reveals an important test-taking insight: when working with sequences, always double-check your position counting by writing out each term's position number. For Fibonacci problems, write the sequence vertically with position numbers clearly labeled. This prevents the most common trap: getting the right calculation method but landing on the wrong term due to miscounting.

Question 11

The sequence $6, 11, 21, 41, 81, ...$ continues with the same pattern. What comes next?

161 (correct answer)
151
171
141

Explanation: When you encounter a sequence problem, your first step is to look for the pattern by examining the differences between consecutive terms. Let's find the differences:

11 - 6 = 5

21 - 11 = 10

41 - 21 = 20

81 - 41 = 40

. The differences are

5, 10, 20, 40

. Notice that each difference doubles:

5 \times 2 = 10

10 \times 2 = 20

20 \times 2 = 40

. Following this pattern, the next difference should be

40 \times 2 = 80

. Therefore, the next term is

81 + 80 = 161

. Looking at the answer choices: Choice A (161) is correct based on our pattern. Choice B (151) would result from adding 70 instead of 80, suggesting someone might have added 30 to the previous difference rather than doubling it. Choice C (171) would come from adding 90, which might result from incorrectly thinking the differences increase by 30 each time (5, 10, 20, 40 doesn't follow this pattern, but 40 + 50 = 90 if someone mistakenly thought differences went up by 10, then 20, then 30). Choice D (141) would result from adding only 60, which doesn't follow any clear mathematical pattern from the given sequence. For sequence problems on the HSPT, always check the differences between terms first. If the first differences don't show an obvious pattern, look at whether they form their own sequence—often involving doubling, tripling, or adding a constant.

Question 12

Consider the sequence $4, 12, 36, 108, 324, ...$ If this pattern continues, what is the 7th term?

2916 (correct answer)
1944
972
2187

Explanation: When you encounter a sequence of numbers, your first step should be identifying the pattern. Look at how each term relates to the previous one to determine if it's arithmetic (adding/subtracting a constant) or geometric (multiplying/dividing by a constant). Let's examine the ratios between consecutive terms:

\frac{12}{4} = 3

\frac{36}{12} = 3

\frac{108}{36} = 3

\frac{324}{108} = 3

. Since each term is exactly 3 times the previous term, this is a geometric sequence with first term

a_1 = 4

and common ratio

r = 3

. For any geometric sequence, the nth term formula is

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

. To find the 7th term:

a_7 = 4 \cdot 3^{(7-1)} = 4 \cdot 3^6 = 4 \cdot 729 = 2916

. Looking at the wrong answers: Choice B (1944) equals

4 \cdot 3^5 \cdot \frac{2}{3}

, suggesting a calculation error where someone might have used the wrong exponent. Choice C (972) equals

4 \cdot 3^5

, which would be the 6th term, indicating someone miscounted the position. Choice D (2187) equals

3^7

, which represents using the wrong first term—someone likely forgot to multiply by the initial value of 4. The correct answer is A) 2916. Strategy tip: For sequence problems, always verify your pattern holds for at least three consecutive terms, then double-check your position counting. Many students lose points by confusing which term number they're actually calculating.

Question 13

In the sequence $2, 3, 5, 9, 17, 33, \ldots$ , what pattern rule generates the next term?

Multiply the previous term by $2$ and subtract $1$ (correct answer)
Add consecutive odd numbers starting with $1$
Double the previous term and add $1$
Add the previous term to itself minus $1$

Explanation: When you encounter a sequence problem, your goal is to identify the consistent pattern that connects each term to the next. Start by examining the differences or relationships between consecutive terms. Let's test each pattern against the given sequence

2, 3, 5, 9, 17, 33, \ldots

Option A suggests multiplying by 2 and subtracting 1. Starting with 2:

2 \times 2 - 1 = 3

. Then

3 \times 2 - 1 = 5

. Next,

5 \times 2 - 1 = 9

. Continuing:

9 \times 2 - 1 = 17

and

17 \times 2 - 1 = 33

. This pattern works perfectly for every term. Option B (adding consecutive odd numbers starting with 1) would give us:

2 + 1 = 3

3 + 3 = 6

6 + 5 = 11

. Since we get 6 instead of 5 for the third term, this pattern fails immediately. Option C (doubling and adding 1) produces:

2 \times 2 + 1 = 5

, but we need the second term to be 3, not 5. This pattern doesn't match from the start. Option D (adding the previous term to itself minus 1) means:

2 + (2-1) = 4

, but our second term is 3, not 4. This also fails immediately. Remember that sequence problems require you to test patterns systematically. Don't assume the first pattern that seems reasonable is correct—verify it works for multiple terms, and always check why the other options fail to avoid similar mistakes in the future.

Question 14

The sequence $1, 8, 27, 64, 125, ...$ follows a clear pattern. What is the next term?

196
216 (correct answer)
225
256

Explanation: When you encounter a sequence like this, look for patterns in how each term relates to its position. The key is recognizing that these numbers represent perfect cubes. Let's examine the pattern:

1 = 1^3

8 = 2^3

27 = 3^3

64 = 4^3

, and

125 = 5^3

. Each term is simply the cube of its position in the sequence. This means the sixth term should be

6^3 = 6 \times 6 \times 6 = 216

. Looking at the wrong answers: Choice A) 196 equals

14^2

, which might tempt you if you mistakenly thought this was a sequence of perfect squares, but

6^2 = 36

, not 196. Choice C) 225 equals

15^2

, another perfect square that doesn't fit our cube pattern. Choice D) 256 equals

4^4

16^2

, which could mislead you if you thought the pattern involved fourth powers or jumped to a different type of sequence entirely. The correct answer is B) 216, since

6^3 = 216

. Study tip: When you see sequence problems, always check if the terms are perfect powers (squares, cubes, fourth powers, etc.) by testing small integers. Recognizing these common mathematical relationships—like

1^3, 2^3, 3^3

—will help you spot patterns quickly. The HSPT often uses sequences based on perfect powers, so memorizing the first few cubes (1, 8, 27, 64, 125, 216) can save you valuable time.

Question 15

Consider the sequence: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots$ . What is the sum of the $8$ th and $9$ th terms?

$\frac{17}{9}$
$\frac{161}{90}$ (correct answer)
$\frac{89}{45}$
$\frac{179}{90}$

Explanation: When you encounter a sequence problem, your first step is identifying the pattern to find the general term formula. Looking at this sequence:

\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots

, notice that each fraction has the form

\frac{n}{n+1}

where

n

represents the term number. The numerator equals the position number, and the denominator is one more than that. So the 8th term is

\frac{8}{9}

and the 9th term is

\frac{9}{10}

. To find their sum, you need a common denominator:

\frac{8}{9} + \frac{9}{10} = \frac{8 \times 10}{9 \times 10} + \frac{9 \times 9}{10 \times 9} = \frac{80}{90} + \frac{81}{90} = \frac{161}{90}

This confirms answer choice B is correct. Let's examine why the other answers are wrong: A)

\frac{17}{9}

results from incorrectly adding

8 + 9 = 17

as the numerator and using just 9 as the denominator, ignoring proper fraction addition rules. C)

\frac{89}{45}

comes from using 45 as the common denominator instead of 90, likely from finding

\text{lcm}(9,10)

incorrectly or making arithmetic errors in the conversion. D)

\frac{179}{90}

suggests calculation errors in the numerator, possibly from incorrect cross-multiplication or adding the converted fractions wrong. Remember: sequence problems require two key steps—identify the pattern to write the general term, then perform careful arithmetic. Always double-check your fraction operations, especially when finding common denominators.

Question 16

In the sequence $5, 8, 14, 26, 50, ...$ each term follows a specific pattern. What is the next term?

86
94
98 (correct answer)
102

Explanation: When you encounter a sequence problem, your first step is to look for the pattern by examining the differences between consecutive terms. Let's find the differences:

8-5=3

14-8=6

26-14=12

50-26=24

. The differences are

3, 6, 12, 24

. Notice that each difference doubles the previous one:

3 \times 2 = 6

6 \times 2 = 12

12 \times 2 = 24

. Following this pattern, the next difference should be

24 \times 2 = 48

. Therefore, the next term is

50 + 48 = 98

. Looking at the wrong answers: Choice A (86) would result from adding 36 to 50, but 36 doesn't fit our doubling pattern. Choice B (94) comes from adding 44, which also breaks the pattern. Choice D (102) results from adding 52, another number that doesn't follow our established rule. These incorrect answers likely come from students who either looked for an arithmetic pattern (constant differences) or noticed the doubling pattern but made calculation errors. Some might have tried to find a pattern in the original sequence itself rather than in the differences. Study tip: For sequence problems on the HSPT, always examine the differences between terms first. If the first differences don't show an obvious pattern, check if the differences themselves follow a pattern—like doubling, adding a constant, or following their own sequence. This systematic approach will help you crack even complex sequences.

Question 17

The sequence $1, 3, 7, 15, 31, 63, ...$ continues with the same pattern. What is the next term?

125
127 (correct answer)
131
135

Explanation: When you encounter a sequence problem, your goal is to identify the pattern that generates each term from the previous one. Look for consistent differences, ratios, or operations between consecutive terms. Let's examine the differences between consecutive terms in this sequence:

$3 - 1 = 2$
$7 - 3 = 4$
$15 - 7 = 8$
$31 - 15 = 16$
$63 - 31 = 32$

The differences are

2, 4, 8, 16, 32

— each difference is doubling! This means the next difference should be

32 \times 2 = 64

. Therefore, the next term is

63 + 64 = 127

. You can also think of this pattern another way: each term equals

2^n - 1

where

n

is the position number. The first term is

2^1 - 1 = 1

, the second is

2^2 - 1 = 3

, and so on. The seventh term would be

2^7 - 1 = 128 - 1 = 127

. Choice A (125) would result from adding 62 instead of 64, missing the doubling pattern. Choice C (131) comes from adding 68, overshooting the correct difference. Choice D (135) results from adding 72, which doesn't follow any recognizable pattern from this sequence. The correct answer is B (127). Study tip: When analyzing sequences, always check if the differences between terms follow a pattern themselves. Powers of 2 appear frequently on standardized tests, so memorize the first several:

2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, 2^6 = 64

Question 18

In the sequence $3, 7, 15, 31, 63, ...$ each term after the first is formed by a specific rule. What is the next term?

127 (correct answer)
125
131
135

Explanation: When you encounter a sequence problem, your first step is to look for the pattern connecting consecutive terms. Don't just look at differences—sometimes the rule involves multiplication, addition of increasing amounts, or other operations. Let's examine how each term relates to the previous one:

From 3 to 7: $3 \times 2 + 1 = 7$
From 7 to 15: $7 \times 2 + 1 = 15$
From 15 to 31: $15 \times 2 + 1 = 31$
From 31 to 63: $31 \times 2 + 1 = 63$

The pattern is clear: multiply each term by 2, then add 1 to get the next term. Applying this rule to find the next term:

63 \times 2 + 1 = 126 + 1 = 127

. Answer A (127) is correct—it follows the established pattern perfectly. Answer B (125) represents what you'd get if you mistakenly subtracted 1 instead of adding 1:

63 \times 2 - 1 = 125

. Answer C (131) might result from incorrectly identifying the pattern as "add 4, then add 8, then add 16..." and continuing with "add 32," but this misses the actual doubling-plus-one rule. Answer D (135) could come from seeing the differences between terms (4, 8, 16, 32) and mistakenly thinking the next difference should be 72 instead of 64. Study tip: In sequence problems, always verify your pattern works for at least three consecutive pairs before applying it. Write out the rule explicitly—this prevents calculation errors and helps you double-check your work.

Question 19

A sequence has the property that the sum of any three consecutive terms is always $30$ . If the first term is $7$ and the second term is $11$ , what is the $8$ th term?

$7$
$11$ (correct answer)
$12$
$9$

Explanation: When you encounter a sequence problem where consecutive terms follow a specific pattern, look for how the constraint creates a cycle that repeats. Given that any three consecutive terms sum to 30, let's call the terms

a_1, a_2, a_3, \ldots

We know

a_1 = 7

and

a_2 = 11

. Since

a_1 + a_2 + a_3 = 30

, we get

7 + 11 + a_3 = 30

, so

a_3 = 12

. Now here's the key insight: if

a_1 + a_2 + a_3 = 30

and

a_2 + a_3 + a_4 = 30

, then these sums are equal. Subtracting the first equation from the second gives us

a_4 - a_1 = 0

, meaning

a_4 = a_1 = 7

. Similarly,

a_5 = a_2 = 11

and

a_6 = a_3 = 12

. The sequence repeats every three terms:

7, 11, 12, 7, 11, 12, \ldots

To find the 8th term, divide 8 by 3:

8 = 3 \times 2 + 2

. The remainder is 2, so

a_8

has the same value as

a_2 = 11

. Choice A (7) would be correct if you mistakenly thought the 8th term corresponded to the first position in the cycle. Choice C (12) represents the third term in the repeating pattern. Choice D (9) doesn't appear anywhere in this sequence and might result from calculation errors. The correct answer is B. Strategy tip: In sequence problems with constraints on consecutive terms, look for repeating patterns. The constraint often forces the sequence to cycle, making long-term predictions manageable through modular arithmetic.

Question 20

The sequence $3, 12, 48, 192, 768, ...$ follows a geometric pattern. What is the 7th term?

12288 (correct answer)
9216
6144
15360

3, 12, 48, 192, 768, ...

Find the ratio:

\frac{12}{3} = 4

\frac{48}{12} = 4

\frac{192}{48} = 4

The common ratio is 4, meaning each term is 4 times the previous term. To find the 7th term, continue the pattern from the 5th term (768):

6th term: $768 \times 4 = 3072$
7th term: $3072 \times 4 = 12288$

You can also use the geometric sequence formula:

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

, where

a_1 = 3

r = 4

, and

n = 7

a_7 = 3 \times 4^6 = 3 \times 4096 = 12288

This confirms answer A is correct. Answer B (9216) would result from incorrectly using

r = 3

instead of

r = 4

. Answer C (6144) comes from miscalculating

4^6

as 2048 instead of 4096. Answer D (15360) might result from using the wrong first term or making an arithmetic error in the multiplication. Strategy tip: Always verify your common ratio by checking it works for multiple consecutive pairs in the sequence. This catches early errors and builds confidence in your calculations.