Identify Number Sequences - HSPT Quantitative

The sequence $3, 12, 48, 192, 768, ...$ follows a geometric pattern. What is the 7th term?
Answer: 12288 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: $3072 \times 4 = 12288$ 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.
1, 2, 6, 15, 31, ?
Answer: 56 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.
In the sequence $2, 9, 23, 44, 72, 107, ...$ what pattern governs the terms, and what is the next value?
Answer: 149 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: - $9 - 2 = 7$ - $23 - 9 = 14$ - $44 - 23 = 21$ - $72 - 44 = 28$ - $107 - 72 = 35$ The first differences are: $7, 14, 21, 28, 35$ . Notice these are all multiples of 7: $7×1, 7×2, 7×3, 7×4, 7×5$ . The pattern shows that each difference increases by 7. Following this pattern, the next difference should be $7×6 = 42$ . Therefore, the next term is $107 + 42 = 149$ . Answer A (149) is correct because it follows the established pattern of differences increasing by 7 each time. Answer B (155) would require a difference of 48, breaking the pattern of multiples of 7. Answer C (142) would need a difference of 35, which we already used for the previous step. Answer D (160) would require a difference of 53, which doesn't fit the sequence of multiples of 7. Strategy tip: For sequence problems, always calculate the first differences between consecutive terms. If those don't show a clear pattern, try second differences (differences of the differences). Most HSPT sequence problems will reveal their pattern within the first or second level of differences, and arithmetic patterns involving consistent increases are especially common.
In the sequence $3, 7, 15, 31, 63, ...$ each term after the first is formed by a specific rule. What is the next term?
Answer: 127 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.
4, 5, 10, 11, 33, 34, ?
Answer: 136 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.
In the sequence $5, 8, 14, 26, 50, ...$ each term follows a specific pattern. What is the next term?
Answer: 98 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.
3, 7, 5, 11, 9, 17, 15, ?
Answer: 25 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.
In the arithmetic sequence $7, 12, 17, 22, 27, ...$ what is the 20th term?
Answer: 102 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.
Consider the sequence $10, 13, 19, 28, 40, 55, ...$ What is the next term?
Answer: 73 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: - $13 - 10 = 3$ - $19 - 13 = 6$ - $28 - 19 = 9$ - $40 - 28 = 12$ - $55 - 40 = 15$ The first differences are: $3, 6, 9, 12, 15, ...$ Notice these increase by 3 each time. This tells us we have a second-order sequence where the differences themselves form an arithmetic pattern. Following this pattern, the next difference should be $15 + 3 = 18$ . Therefore, the next term is $55 + 18 = 73$ . Looking at the answer choices: A) 73 is correct based on our pattern analysis. B) 75 might result from incorrectly adding 20 instead of 18, perhaps by misidentifying the pattern in differences. C) 70 would come from adding only 15 (the previous difference) without recognizing that the differences themselves increase. D) 78 likely results from adding 23, which might occur if you mistakenly think the differences increase by 5 instead of 3. For sequence problems on the HSPT, always check both first and second differences. Many sequences that don't follow simple arithmetic or geometric patterns will reveal their structure through second differences. This systematic approach will help you avoid the common trap of assuming the most recent difference continues unchanged.
In the sequence $0, 1, 5, 14, 30, 55, ...$ what is the underlying pattern, and what comes next?
Answer: 91 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.
Consider the sequence $1, 4, 9, 16, 25, 36, ...$ What is the 15th term?
Answer: 225 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.
The sequence $1, 3, 7, 15, 31, 63, ...$ continues with the same pattern. What is the next term?
Answer: 127 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: $2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, 2^6 = 64$ .
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?
Answer: $\frac{161}{90}$ 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.
The sequence $4, 7, 13, 25, 49, ...$ follows a specific rule. What is the 6th term?
Answer: 97 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: - $7 - 4 = 3$ - $13 - 7 = 6$ - $25 - 13 = 12$ - $49 - 25 = 24$ The differences are $3, 6, 12, 24$ . Notice that each difference doubles the previous one: $3 \times 2 = 6$ , $6 \times 2 = 12$ , $12 \times 2 = 24$ . This means the next difference should be $24 \times 2 = 48$ . Therefore, the 6th term is $49 + 48 = 97$ . Now let's examine why the other answers are incorrect. Answer B (89) would result from adding 40 to 49, but there's no pattern that gives us 40 as the next difference. Answer C (101) comes from adding 52 to 49, which also doesn't follow our doubling pattern. Answer D (105) results from adding 56 to 49, again breaking the established rule. The correct answer is A (97) because it follows the pattern where differences double each time. Study tip: For sequence problems, always look at the differences between terms first. If the first differences don't show a clear pattern, check if the differences themselves follow a pattern (like doubling, as in this problem). This two-step approach will help you crack most sequence questions on the HSPT.
In the sequence $1, 5, 14, 30, 55, 91, ...$ what pattern emerges, and what is the next term?
Answer: 140 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: - $5 - 1 = 4$ - $14 - 5 = 9$ - $30 - 14 = 16$ - $55 - 30 = 25$ - $91 - 55 = 36$ The differences are $4, 9, 16, 25, 36$ . Notice these are perfect squares: $2^2, 3^2, 4^2, 5^2, 6^2$ . This reveals the pattern: each term increases by the square of consecutive integers starting with 2. Following this pattern, the next difference should be $7^2 = 49$ . Therefore, the next term is $91 + 49 = 140$ . Choice A (140) is correct because it follows the established pattern of adding consecutive perfect squares. Choice B (132) would result from adding 41 to the previous term, which doesn't fit the perfect square pattern. Choice C (148) would come from adding 57, suggesting someone might have incorrectly identified a different pattern or made calculation errors. Choice D (135) represents adding 44, which also breaks the perfect square sequence we established. Study tip: When analyzing sequences, always check multiple levels of differences. If first differences don't show an obvious pattern, examine second differences or look for relationships involving squares, cubes, or other mathematical operations. Sequential perfect squares are particularly common on standardized tests.
In the sequence $3, 9, 21, 45, 93, ...$ what rule generates each term, and what comes next?
Answer: 189 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: - $9 - 3 = 6$ - $21 - 9 = 12$ - $45 - 21 = 24$ - $93 - 45 = 48$ The differences are $6, 12, 24, 48, ...$ Notice that each difference doubles: $6 \times 2 = 12$ , $12 \times 2 = 24$ , $24 \times 2 = 48$ . This means the next difference should be $48 \times 2 = 96$ . Therefore, the next term is $93 + 96 = 189$ . You can also express this pattern as: each term equals double the previous term plus a power of 3. For example: $9 = 2(3) + 3$ , $21 = 2(9) + 3$ , $45 = 2(21) + 3$ , $93 = 2(45) + 3$ . So the next term is $2(93) + 3 = 189$ . Choice A (189) is correct. Choice B (186) might result from miscalculating $93 + 96$ or using an incorrect difference of 93. Choice C (201) could come from incorrectly assuming the differences increase by addition rather than multiplication. Choice D (195) might arise from using the wrong doubling pattern or arithmetic errors. Study tip: For sequence problems, always examine the differences between terms first. If those differences don't form an obvious pattern, look at second differences or try to spot multiplication/addition relationships between consecutive terms.
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?
Answer: $144$ 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.
64, 32, 36, 18, 22, ?
Answer: 11 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.
The sequence $7, 14, 28, 56, 112, ...$ continues with the same rule. What is the 8th term?
Answer: 896 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. Examining this sequence: $7, 14, 28, 56, 112, ...$ Notice that $14 = 7 \times 2$ , $28 = 14 \times 2$ , $56 = 28 \times 2$ , and $112 = 56 \times 2$ . The rule is multiplication by 2 each time—this is a geometric sequence with first term $a_1 = 7$ and common ratio $r = 2$ . To find the 8th term, continue the pattern: - 6th term: $112 \times 2 = 224$ - 7th term: $224 \times 2 = 448$ - 8th term: $448 \times 2 = 896$ Therefore, the 8th term is 896, which is answer choice A. Looking at the wrong answers: B) 672 doesn't follow any clear mathematical relationship to the established pattern. C) 448 is actually the 7th term—this catches students who miscount or stop one step early. D) 1344 would be the 9th term ( $896 \times 2$ ), trapping those who count one position too far. When working with sequences, always double-check your term counting by writing out the position numbers above each term. Geometric sequences appear frequently on standardized tests, so remember the key formula: $a_n = a_1 \times r^{n-1}$ , where $a_1$ is the first term and $r$ is the common ratio.
In the sequence $5, 15, 45, 135, 405, ...$ each term follows the same rule. What is the 7th term?
Answer: 3645 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: - $15 ÷ 5 = 3$ - $45 ÷ 15 = 3$ - $135 ÷ 45 = 3$ - $405 ÷ 135 = 3$ The rule is clear: each term is obtained by multiplying the previous term by 3. This is a geometric sequence with first term $a_1 = 5$ and common ratio $r = 3$ . To find the 7th term, continue the pattern: - 5th term: $405$ - 6th term: $405 × 3 = 1215$ - 7th term: $1215 × 3 = 3645$ Therefore, the answer is A) 3645. Let's examine why the other choices are incorrect: - 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. Strategy tip: For sequence problems, always verify your pattern works for at least three consecutive pairs before applying it. Write out each step clearly to avoid counting errors, and double-check your arithmetic since these problems often include answer choices that result from common calculation mistakes.

Start this quiz