Pattern Rules From Sequences - SHSAT Math

Consider the sequence: $5, 8, 14, 26, 50, ...$ If $S_n$ represents the $n$ th term, which of the following best describes the pattern rule?
Answer: $S_n = S_{n-1} + 3 \cdot 2^{n-2}$ for $n \geq 2$ Look at the differences between consecutive terms: 8 − 5 = 3, 14 − 8 = 6, 26 − 14 = 12 and 50 − 26 = 24. Each difference is double the one before, so the difference added to get term $n$ is $3 \cdot 2^{n-2}$ : 3 for $n = 2$ , 6 for $n = 3$ , 12 for $n = 4$ , and so on. That gives $S_n = S_{n-1} + 3 \cdot 2^{n-2}$ . Check: 5 + 3 = 8, 8 + 6 = 14, 14 + 12 = 26, 26 + 24 = 50. The rule $S_n = 2S_{n-1} - 1$ gives 2(5) − 1 = 9 as the second term, not 8. The rule $S_n = S_{n-1} + 2^{n-1}$ adds 2, then 4, giving 5, 7, 11. The rule $S_n = S_{n-1} + 2^n - 1$ adds 3 and then 7, giving 5, 8, 15, so it matches only the first step.
A sequence begins $1, 4, 9, 16, 25, 36, ...$ and another begins $1, 3, 6, 10, 15, 21, ...$ If $T_n$ represents the $n$ th term when these sequences are combined by taking their sum term by term, which expression represents $T_n$ ?
Answer: $n^2 + \frac{n(n+1)}{2}$ The first sequence is 1², 2², 3², 4², ... = n². The second sequence is triangular numbers: 1, 1+2, 1+2+3, 1+2+3+4, ... = n(n+1)/2. Therefore, Tₙ = n² + n(n+1)/2. Let's verify: T₁ = 1 + 1 = 2, T₂ = 4 + 3 = 7, T₃ = 9 + 6 = 15. Using option C: T₁ = 1 + 1 = 2 ✓, T₂ = 4 + 3 = 7 ✓, T₃ = 9 + 6 = 15 ✓.
Identify the rule for the alternating sequence $9,\;16,\;8,\;15,\;7,\;14,\ldots$ .
Answer: Add 7, then subtract 8, repeating that two-step cycle. When you encounter an alternating sequence like this, you need to identify the pattern by examining how each term relates to the ones around it. Look at the differences between consecutive terms to spot the repeating cycle. Let's trace through the sequence step by step: $9,\;16,\;8,\;15,\;7,\;14,\ldots$ From 9 to 16: $16 - 9 = +7$ From 16 to 8: $8 - 16 = -8$ From 8 to 15: $15 - 8 = +7$ From 15 to 7: $7 - 15 = -8$ From 7 to 14: $14 - 7 = +7$ The pattern is clear: add 7, subtract 8, add 7, subtract 8, and so on. This confirms that choice A is correct. Now let's see why the other options fail. Choice B suggests subtracting 1, then adding 7. But $9 - 1 = 8$ , not 16, so this doesn't work from the very first step. Choice C proposes adding 7 twice, then subtracting 8 once. Following this rule: $9 + 7 = 16$ , $16 + 7 = 23$ (but we need 8), so this pattern breaks immediately. Choice D claims we subtract 7 each time, but $9 - 7 = 2$ , not 16, so this is clearly wrong. Strategy tip: For alternating sequences, always check at least 4-5 terms to confirm the pattern repeats correctly. Write out the differences between consecutive terms—this makes the underlying rule much easier to spot than trying to see it in your head.
The terms $\tfrac12,\;-1,\;2,\;-4,\;8,\ldots$ follow what pattern rule?
Answer: Multiply each term by −2 to obtain the next term. When you encounter a sequence problem, your goal is to identify the consistent rule that transforms each term into the next one. Look at how each term relates to the previous term. Let's examine the pattern by checking what operation transforms each term to the next: - From $\frac{1}{2}$ to $-1$ : $\frac{1}{2} \times (-2) = -1$ ✓ - From $-1$ to $2$ : $(-1) \times (-2) = 2$ ✓ - From $2$ to $-4$ : $2 \times (-2) = -4$ ✓ - From $-4$ to $8$ : $(-4) \times (-2) = 8$ ✓ Every term multiplied by $-2$ gives the next term, confirming that choice A is correct. Now let's see why the other options fail. Choice B suggests adding 1.5 repeatedly, but $\frac{1}{2} + 1.5 = 2$ , not $-1$ . Choice C proposes alternating between adding 1 and subtracting 3, but $\frac{1}{2} + 1 = 1.5$ , not $-1$ . Choice D suggests dividing by $-2$ , but $\frac{1}{2} \div (-2) = -\frac{1}{4}$ , not $-1$ . For sequence problems on the SHSAT, always test your suspected pattern on multiple consecutive terms—don't just check the first pair. A true pattern rule must work for every single transition in the sequence. Start by looking for multiplication or division patterns first, as they're common in geometric sequences like this one.
A sequence begins: $3, 7, 15, 31, 63, ...$ Each term after the first is generated by applying the same rule to the previous term. What is the 8th term of this sequence?
Answer: 511 The pattern rule is: multiply by 2 and add 1. Starting with 3: 3×2+1=7, 7×2+1=15, 15×2+1=31, 31×2+1=63, 63×2+1=127 (6th term), 127×2+1=255 (7th term), 255×2+1=511 (8th term). Therefore, the 8th term is 511.
Examine $2,\;3,\;5,\;8,\;12,\;17,\ldots$ . Which statement describes the pattern of successive differences that generates the sequence?
Answer: Add consecutive whole numbers starting with 1 (1, 2, 3, 4, …). When you encounter a sequence problem, look for patterns in how the terms change from one to the next. The key is examining the differences between consecutive terms to understand the underlying rule. Let's find the differences between consecutive terms in this sequence: $2, 3, 5, 8, 12, 17, \ldots$ - From 2 to 3: difference is 1 - From 3 to 5: difference is 2 - From 5 to 8: difference is 3 - From 8 to 12: difference is 4 - From 12 to 17: difference is 5 The pattern of differences is $1, 2, 3, 4, 5, \ldots$ — consecutive whole numbers starting with 1. This confirms answer choice A is correct. Let's examine why the other options fail. Choice B suggests adding consecutive even numbers (2, 4, 6, 8...), but our differences are 1, 2, 3, 4, 5 — not even numbers. Choice C proposes adding consecutive primes starting with 3 (3, 5, 7, 11...), but again, our actual differences don't match this pattern. Choice D describes a multiplication rule rather than an addition pattern, and applying "multiply by 2 then subtract 1" to 2 gives 3, then to 3 gives 5, but to 5 gives 9 (not 8), so this rule breaks down immediately. Study tip: For sequence problems, always calculate the first few differences between consecutive terms before looking at the answer choices. This prevents you from being misled by attractive but incorrect patterns and helps you identify the true underlying rule quickly.
Which description fits the sequence $7,\;10,\;15,\;22,\;31,\ldots$ ?
Answer: Add consecutive odd numbers starting with 3 (3, 5, 7, 9, …). When you encounter a sequence problem, your goal is to identify the pattern by examining how each term relates to the previous one. Look at the differences between consecutive terms to reveal the underlying rule. Let's analyze the differences in this sequence: $7, 10, 15, 22, 31, \ldots$ From 7 to 10: difference of 3 From 10 to 15: difference of 5 From 15 to 22: difference of 7 From 22 to 31: difference of 9 The differences are 3, 5, 7, 9 — consecutive odd numbers! This confirms that choice A is correct: you add consecutive odd numbers starting with 3. Now let's see why the other options fail: Choice B suggests adding 3 repeatedly. If this were true, the sequence would be $7, 10, 13, 16, 19, \ldots$ — clearly different from our actual sequence. Choice C says "multiply by 1 then add 3," which is just adding 3 (since multiplying by 1 doesn't change anything). This produces the same incorrect sequence as choice B. Choice D claims we alternate between adding 3 and adding 5. Following this pattern: $7 + 3 = 10$ , $10 + 5 = 15$ , $15 + 3 = 18$ . But our sequence shows 22, not 18, so this pattern breaks down immediately. Study tip: For sequence problems, always write out the differences between consecutive terms. This often reveals patterns that aren't obvious from looking at the terms themselves. Many SHSAT sequence questions test whether you can spot patterns in these differences rather than in the original numbers.
A recursive sequence is defined where $a_1 = 2$ and $a_n = 3a_{n-1} - 1$ for $n \geq 2$ . What is $a_6 - a_5$ ?
Answer: 243 Computing the sequence: a₁=2, a₂=3(2)-1=5, a₃=3(5)-1=14, a₄=3(14)-1=41, a₅=3(41)-1=122, a₆=3(122)-1=365. Therefore a₆-a₅=365-122=243. Note that the differences between consecutive terms form powers of 3: a₂-a₁=3, a₃-a₂=9, a₄-a₃=27, a₅-a₄=81, a₆-a₅=243.
Given the sequence $2,\;5,\;10,\;17,\;26,\ldots$ , what rule produces each new term?
Answer: Add consecutive odd numbers beginning with 3 to each term. When you encounter sequence problems, look for patterns in how terms change from one to the next. Calculate the differences between consecutive terms to reveal the underlying rule. Let's examine the differences: $5-2=3$ , $10-5=5$ , $17-10=7$ , $26-17=9$ . The differences are $3, 5, 7, 9$ – consecutive odd numbers! This means we add 3 to get the second term, add 5 to get the third term, add 7 to get the fourth term, and so on. Choice A correctly describes this pattern: "Add consecutive odd numbers beginning with 3 to each term." Let's verify why the other choices fail. Choice B suggests multiplying by 2 and subtracting 1. Testing: $2 \times 2 - 1 = 3$ , not 5, so this rule breaks immediately. Choice C proposes adding 3 repeatedly, which would give us $2, 5, 8, 11, 14...$ – clearly different from our sequence. Choice D suggests alternating between adding 2 and 3, producing $2, 4, 7, 9, 12...$ or $2, 5, 7, 10, 12...$ depending on which you start with – neither matches our sequence. For sequence problems on the SHSAT, always calculate the differences between consecutive terms first. If those differences don't form an obvious pattern, try looking at second differences (differences of differences) or ratios. Most sequence problems test your ability to spot arithmetic progressions, geometric progressions, or patterns like this one where the differences themselves follow a rule.
A sequence is $1,\;1,\;2,\;3,\;5,\;8,\ldots$ . What rule produces each term after the second?
Answer: Add the two preceding terms to get the next term. When you encounter a sequence problem, your goal is to identify the pattern that generates each term. Look at how consecutive terms relate to each other, and test whether that pattern holds throughout the sequence. Let's examine this sequence: $1, 1, 2, 3, 5, 8, \ldots$ Starting from the third term, let's see what rule produces each number: - Third term: $1 + 1 = 2$ ✓ - Fourth term: $1 + 2 = 3$ ✓ - Fifth term: $2 + 3 = 5$ ✓ - Sixth term: $3 + 5 = 8$ ✓ This confirms that each term equals the sum of the two preceding terms. This is the famous Fibonacci sequence. Now let's check why the other options fail: Option B suggests doubling the previous term then subtracting 1. Testing: $2(1) - 1 = 1$ , but the third term is 2, not 1. Option C proposes alternating between adding 1 and multiplying by 2. Starting from the second term: $1 + 1 = 2$ (add 1), then $2 \times 2 = 4$ (multiply by 2). But the fourth term is 3, not 4. Option D suggests squaring the position number. For position 3: $3^2 = 9$ , but the third term is 2, not 9. Study tip: For sequence problems, always test the proposed rule on at least three consecutive terms. Many wrong answers will work for one or two terms but break down quickly. The Fibonacci sequence appears frequently on standardized tests, so recognizing this "sum of previous two terms" pattern will save you time.
Which single operation generates the sequence $120,\;60,\;30,\;15,\;7.5,\ldots$ ?
Answer: Divide the previous term by 2. When you encounter a sequence problem, your goal is to identify the pattern by examining how each term relates to the previous one. Look at the relationship between consecutive terms to find the single operation that generates the entire sequence. Let's examine each transition in the sequence $120, 60, 30, 15, 7.5, \ldots$ : - From 120 to 60: $120 ÷ 2 = 60$ - From 60 to 30: $60 ÷ 2 = 30$ - From 30 to 15: $30 ÷ 2 = 15$ - From 15 to 7.5: $15 ÷ 2 = 7.5$ Each term is exactly half the previous term, confirming that choice A is correct. Now let's check why the other options fail. Choice B suggests subtracting 60 each time, but $60 - 60 = 0$ , not 30, so this doesn't work beyond the first step. Choice C proposes multiplying by 0.25 repeatedly. While $120 × 0.25 = 30$ , this skips the term 60 entirely and doesn't match the given sequence. Choice D suggests alternating between dividing by 2 and by 3. If we tried this: $120 ÷ 2 = 60$ , then $60 ÷ 3 = 20$ , but our sequence shows 30, not 20. When working with sequences on the SHSAT, always test your identified pattern against multiple consecutive terms. A single operation should work consistently throughout the entire sequence. Don't just check the first transition—verify that your pattern holds for at least three or four terms to avoid falling into trap answers.
Consider the pattern $5,\;8,\;4,\;7,\;3,\;6,\ldots$ . What is the rule?
Answer: Add 3, then subtract 4, and repeat this two-step process. When you encounter a sequence problem, you need to identify the pattern by examining how each term relates to the previous one. Look at the differences or operations between consecutive terms to find the underlying rule. Let's trace through this sequence: $5, 8, 4, 7, 3, 6, \ldots$ From 5 to 8: we add 3 From 8 to 4: we subtract 4 From 4 to 7: we add 3 From 7 to 3: we subtract 4 From 3 to 6: we add 3 The pattern is clearly: add 3, subtract 4, add 3, subtract 4, and so on. This confirms that choice A is correct. Let's check why the other options fail: Choice B suggests subtracting 1 then doubling. Testing: $5 - 1 = 4$ , then $4 \times 2 = 8$ ✓. But continuing: $8 - 1 = 7$ , then $7 \times 2 = 14 \neq 4$ . This breaks immediately. Choice C claims we add 3 successively. While $5 + 3 = 8$ works, $8 + 3 = 11 \neq 4$ . This fails on the second step. Choice D proposes multiplying by -1 then adding 13. Testing: $5 \times (-1) + 13 = 8$ ✓, but $8 \times (-1) + 13 = 5 \neq 4$ . This also fails quickly. Strategy tip: For sequence problems, always test each proposed rule on at least the first three terms. Many incorrect choices will work for just one step but break down quickly. Write out the operations explicitly rather than trying to do them mentally—this prevents calculation errors that could lead you to the wrong answer.
Which rule describes the pattern $81,\;27,\;9,\;3,\;1,\ldots$ ?
Answer: Divide the previous term by 3 each time. When you encounter a sequence of numbers, your goal is to identify the relationship between consecutive terms. Look at how each term relates to the one before it. Let's examine the pattern in $81, 27, 9, 3, 1, \ldots$ by checking what operation transforms each term into the next: - $81 \div 3 = 27$ - $27 \div 3 = 9$ - $9 \div 3 = 3$ - $3 \div 3 = 1$ Each term is consistently divided by 3 to get the next term. This confirms that choice A is correct. Now let's see why the other options fail. Choice B suggests subtracting 54 each time, but $81 - 54 = 27$ works for the first step, while $27 - 54 = -27$ , which doesn't match the actual second term of 9. Choice C proposes multiplying by −3, but $81 \times (-3) = -243$ , not 27, so this fails immediately. Choice D suggests squaring the reciprocal: $\left(\frac{1}{81}\right)^2 = \frac{1}{6561}$ , which is nowhere close to 27. The key strategy for sequence problems is to test the relationship systematically with the first few terms. Don't just check whether a rule works for one pair—verify it holds throughout the given sequence. Also, notice that this sequence represents powers of 3 in reverse: $3^4, 3^3, 3^2, 3^1, 3^0$ . Recognizing exponential patterns can help you spot division or multiplication rules quickly.
Consider the sequence $3,\;7,\;11,\;15,\;19,\ldots$ . Which rule best describes how each term is generated from the preceding one?
Answer: Add 4 to the previous term. When you encounter a sequence problem, you need to identify the pattern that connects consecutive terms. Look at how each term relates to the one before it by examining the differences or ratios between terms. Let's analyze this sequence: $3, 7, 11, 15, 19, \ldots$ First, find the difference between consecutive terms: - $7 - 3 = 4$ - $11 - 7 = 4$ - $15 - 11 = 4$ - $19 - 15 = 4$ The difference is consistently 4, which means each term is generated by adding 4 to the previous term. This confirms that choice A is correct. Now let's check why the other options fail: Choice B suggests multiplying by 2 and subtracting 1. Testing this: $3 \times 2 - 1 = 5$ , but the second term is 7, not 5. Choice C claims we add consecutive prime numbers starting with 3. The consecutive primes are 3, 5, 7, 11, 13... So we'd have: $3 + 3 = 6$ (not 7), then $6 + 5 = 11$ . This doesn't match our sequence. Choice D suggests multiplying by 4 and dividing by 3. Testing: $3 \times 4 ÷ 3 = 4$ , but our second term is 7. Study tip: For sequence problems, always start by finding the difference between consecutive terms. If the differences are constant, you have an arithmetic sequence where each term equals the previous term plus that constant difference. This is the most common type of sequence on standardized tests.
The sequence $4,\;9,\;16,\;25,\;36,\ldots$ follows which pattern rule?
Answer: Each term is the square of consecutive integers starting with 2. When you encounter a sequence of numbers, your first step should be to look for patterns by examining how each term relates to its position or to previous terms. Let's analyze this sequence systematically. Looking at the given sequence $4, 9, 16, 25, 36, \ldots$ , notice that these are all perfect squares. Specifically: $4 = 2^2$ , $9 = 3^2$ , $16 = 4^2$ , $25 = 5^2$ , and $36 = 6^2$ . The pattern shows each term is the square of consecutive integers starting with 2, making choice A correct. Let's examine why the other options fail. Choice B claims each term increases by 5 consistently. However, $9 - 4 = 5$ , $16 - 9 = 7$ , $25 - 16 = 9$ , and $36 - 25 = 11$ . The differences aren't constant—they increase by 2 each time. Choice C suggests each term equals twice the previous term plus 1. Testing this: $2(4) + 1 = 9$ ✓, but $2(9) + 1 = 19 \neq 16$ ✗. Choice D proposes each term is the cube of its position number. The first term (position 1) would be $1^3 = 1$ , but we have 4 instead. Strategy tip: When analyzing number sequences, first check if the terms are perfect squares, cubes, or other powers. Perfect squares appear frequently on the SHSAT, and recognizing them quickly (like $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$ ) will save you valuable time.
Two students are analyzing the sequence $\frac{1}{1}, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}, ...$ Student A claims the pattern rule is $a_n = \frac{1}{n^2}$ . Student B claims the pattern rule is $a_n = \frac{(-1)^{n+1}}{n^2}$ for odd $n$ only. Which statement is correct?
Answer: Only Student A is correct; Student B's rule doesn't apply to this sequence When analyzing mathematical sequences, you need to carefully examine what the given terms actually show before evaluating proposed formulas. Look at each term in the sequence and test whether the proposed rules generate those exact values. Let's check Student A's formula $a_n = \frac{1}{n^2}$ against the given sequence: - $a_1 = \frac{1}{1^2} = \frac{1}{1}$ ✓ - $a_2 = \frac{1}{2^2} = \frac{1}{4}$ ✓ - $a_3 = \frac{1}{3^2} = \frac{1}{9}$ ✓ - $a_4 = \frac{1}{4^2} = \frac{1}{16}$ ✓ - $a_5 = \frac{1}{5^2} = \frac{1}{25}$ ✓ Student A's formula perfectly matches every term shown. Now examine Student B's claim about $a_n = \frac{(-1)^{n+1}}{n^2}$ for odd $n$ only. This formula would give $a_1 = \frac{1}{1} = 1$ , $a_3 = \frac{1}{9}$ , $a_5 = \frac{1}{25}$ , etc., but provides no rule for the even-positioned terms like $\frac{1}{4}$ and $\frac{1}{16}$ . Since Student B's rule doesn't account for all the terms in the sequence, it's incomplete. Choice A is correct because Student A provides a complete, accurate formula while Student B's rule fails to address the entire sequence. Choice B is wrong because Student B's rule is incomplete. Choice C incorrectly suggests there's an alternating pattern when all terms are clearly positive. Choice D is wrong because no factorials appear in this sequence. Strategy tip: Always test proposed formulas against all given terms, not just a few. A correct sequence formula must account for every single term shown.

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