HSPT Quantitative Quiz: Identify Number Sequences

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.

Question 1

1. $7$
2. $11$ (correct answer)
3. $12$
4. $9$

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 2

1. 12288 (correct answer)
2. 9216
3. 6144
4. 15360

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. Looking at the given sequence: $3, 12, 48, 192, 768, ...$ 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: $3072 \times 4 = 12288$

Question 3

1. 210
2. 225 (correct answer)
3. 240
4. 256

Explanation: When you see a sequence of numbers, your first step is to identify the pattern. Looking at this sequence: $1, 4, 9, 16, 25, 36, ...$ you should recognize these as perfect squares. Notice that $1 = 1^2$, $4 = 2^2$, $9 = 3^2$, $16 = 4^2$, $25 = 5^2$, and $36 = 6^2$. This means the $n$th term of the sequence is simply $n^2$. To find the 15th term, you calculate $15^2 = 15 \times 15 = 225$. Let's examine why the other answers are wrong. Choice A (210) might tempt you if you mistakenly think this is an arithmetic sequence and try to find a common difference, but there's no consistent difference between consecutive terms. Choice C (240) could result from calculation errors when squaring 15, perhaps confusing it with $15 \times 16$. Choice D (256) is actually $16^2$, so you'd get this if you miscounted and thought you needed the 16th term instead of the 15th. The key strategy for sequence problems is pattern recognition. Perfect squares appear frequently on standardized tests, so memorizing the first 15-20 perfect squares will save you time. When you see $1, 4, 9, 16...$ immediately think "perfect squares" rather than trying to find differences between terms. This recognition allows you to jump straight to the formula $n^2$ and solve quickly.

Question 4

1. 86
2. 94
3. 98 (correct answer)
4. 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 5

1. 55 (correct answer)
2. 89
3. 34
4. 21

Explanation: When you encounter a sequence problem, your first step is identifying the pattern or rule that generates each term. This particular sequence is the famous Fibonacci sequence, where each term equals the sum of the two preceding terms. Let's trace through the pattern: $1 + 1 = 2$, $1 + 2 = 3$, $2 + 3 = 5$, $3 + 5 = 8$, $5 + 8 = 13$. Continuing this rule: the 8th term is $8 + 13 = 21$, the 9th term is $13 + 21 = 34$, and the 10th term is $21 + 34 = 55$. Looking at the answer choices, A) 55 is correct as our calculation confirms. B) 89 would actually be the 11th term ($34 + 55 = 89$), representing a common error of miscounting positions in the sequence. C) 34 is the 9th term, showing another off-by-one counting mistake. D) 21 is the 8th term, indicating the student stopped calculating too early or miscounted by two positions. The key trap in Fibonacci problems is losing track of which position you're calculating. Always write out the sequence systematically and double-check your position count. Remember that sequence problems on the HSPT often test your ability to recognize patterns and perform careful, sequential calculations rather than complex mathematical concepts.

Question 6

1. 95 (correct answer)
2. 89
3. 101
4. 91

Explanation: When you encounter a sequence problem, your first step is to identify the pattern by examining how each term relates to the previous ones. Look for differences, ratios, or more complex relationships. Let's find the pattern in this sequence: $2, 5, 11, 23, 47, ...$ First, check the differences between consecutive terms:

• $5 - 2 = 3$
• $11 - 5 = 6$
• $23 - 11 = 12$
• $47 - 23 = 24$

Question 7

1. 102 (correct answer)
2. 97
3. 107
4. 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 8

1. 189 (correct answer)
2. 186
3. 201
4. 195

Explanation: When you encounter a sequence problem, your goal is to identify the pattern that transforms each term into the next. Start by examining the differences between consecutive terms. Looking at the sequence $3, 9, 21, 45, 93, ...$, let's find the differences:

• $9 - 3 = 6$
• $21 - 9 = 12$
• $45 - 21 = 24$
• $93 - 45 = 48$

Question 9

1. 3645 (correct answer)
2. 1215
3. 2430
4. 4860

Explanation: When you encounter a sequence problem, your first step is identifying the pattern or rule that connects consecutive terms. Look at how each term relates to the previous one. Let's examine the given sequence: $5, 15, 45, 135, 405, ...$ To find the pattern, divide each term by the previous term:

• $15 ÷ 5 = 3$
• $45 ÷ 15 = 3$
• $135 ÷ 45 = 3$
• $405 ÷ 135 = 3$

• 5th term: $405$
• 6th term: $405 × 3 = 1215$
• 7th term: $1215 × 3 = 3645$

• B) 1215 is actually the 6th term, not the 7th. This catches students who miscounted.
• C) 2430 equals $405 × 6$, suggesting someone incorrectly thought the rule was "multiply by 6" instead of recognizing the consistent ×3 pattern.
• D) 4860 equals $1215 × 4$, indicating confusion about the multiplier or arithmetic errors in the calculation.

Question 10

1. 73 (correct answer)
2. 75
3. 70
4. 78

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 between each pair of terms:

• $13 - 10 = 3$
• $19 - 13 = 6$
• $28 - 19 = 9$
• $40 - 28 = 12$
• $55 - 40 = 15$

Question 11

1. 161 (correct answer)
2. 151
3. 171
4. 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

1. 128 (correct answer)
2. 112
3. 144
4. 136

Explanation: When you encounter a sequence problem, your first step is to identify the pattern by examining the differences between consecutive terms. Let's find the first differences: $8-2=6$, $18-8=10$, $32-18=14$, $50-32=18$, $72-50=22$. The first differences are $6, 10, 14, 18, 22$. Since these aren't constant, let's find the second differences: $10-6=4$, $14-10=4$, $18-14=4$, $22-18=4$. The second differences are constant at 4, indicating this is a quadratic sequence. Continuing the pattern of first differences: the next difference would be $22+4=26$, then $26+4=30$, then $30+4=34$. Now we can find the remaining terms:

• 6th term: $72+26=98$
• 7th term: $98+30=128$
• 8th term: $128+34=162$

Question 13

1. 91 (correct answer)
2. 85
3. 95
4. 100

Explanation: When you encounter a sequence problem, your first step is to look for patterns in the differences between consecutive terms. This approach often reveals the underlying structure more clearly than trying to spot a direct pattern in the sequence itself. Let's examine the differences between consecutive terms: $1-0=1$, $5-1=4$, $14-5=9$, $30-14=16$, $55-30=25$. The first differences are $1, 4, 9, 16, 25$, which are perfect squares: $1^2, 2^2, 3^2, 4^2, 5^2$. This means the next difference should be $6^2 = 36$, so the next term is $55 + 36 = 91$. You can verify this pattern by recognizing that each term equals the sum of the first $n$ perfect squares. For instance, the fifth term is $30 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16$. Answer A (91) is correct because it follows this established pattern. Answer B (85) would require a difference of 30, which doesn't fit the square pattern. Answer C (95) would need a difference of 40, and answer D (100) would need 45 – neither corresponds to $6^2 = 36$. For sequence questions on the HSPT, always examine the differences between terms when the direct pattern isn't obvious. Many sequences are built on arithmetic or geometric progressions hidden in their differences, and recognizing common number patterns like perfect squares, triangular numbers, or factorials will help you solve these problems efficiently.

Question 14

1. 149 (correct answer)
2. 155
3. 142
4. 160

Explanation: When you encounter a sequence problem, your first step is to look for patterns in the differences between consecutive terms. This systematic approach will reveal the underlying structure. Let's examine the differences between consecutive terms in this sequence:

• $9 - 2 = 7$
• $23 - 9 = 14$
• $44 - 23 = 21$
• $72 - 44 = 28$
• $107 - 72 = 35$

Question 15

1. 1458
2. 1944
3. 4374 (correct answer)
4. 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 16

1. 140 (correct answer)
2. 132
3. 148
4. 135

Explanation: When you encounter a number sequence problem, your goal is to identify the underlying pattern by examining how terms relate to each other. Start by looking at the differences between consecutive terms. For the sequence $1, 5, 14, 30, 55, 91, ...$, let's find the first differences:

• $5 - 1 = 4$
• $14 - 5 = 9$
• $30 - 14 = 16$
• $55 - 30 = 25$
• $91 - 55 = 36$

Question 17

1. 1152 (correct answer)
2. 864
3. 576
4. 2304

Explanation: When you encounter a sequence problem, your first step is 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). In this sequence, examine the ratios: $\frac{18}{9} = 2$, $\frac{36}{18} = 2$, $\frac{72}{36} = 2$, $\frac{144}{72} = 2$. Each term is exactly double the previous term, making this a geometric sequence with a common ratio of 2. To find the 8th term, continue multiplying by 2:

• 5th term: $144 \times 2 = 288$
• 6th term: $288 \times 2 = 576$
• 7th term: $576 \times 2 = 1152$
• 8th term: $1152 \times 2 = 2304$

• 6th term: $144 \times 2 = 288$
• 7th term: $288 \times 2 = 576$
• 8th term: $576 \times 2 = 1152$

Question 18

1. 10 (correct answer)
2. 8
3. 12
4. 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:

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

Question 19

1. 97 (correct answer)
2. 89
3. 101
4. 105

Explanation: When you encounter a sequence problem, your first task is to identify the pattern by examining the differences between consecutive terms. Let's find the differences between each pair of consecutive terms:

• $7 - 4 = 3$
• $13 - 7 = 6$
• $25 - 13 = 12$
• $49 - 25 = 24$

Question 20

1. 2916 (correct answer)
2. 1944
3. 972
4. 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.