Pattern Rules From Sequences - SSAT Middle Level Quantitative

In a sequence, the first term is 7, the second term is 10, and each subsequent term is the average of all the previous terms. What is the 4th term?
Answer: 8.5 When you encounter sequence problems where each term depends on previous terms, you need to calculate step-by-step, carefully tracking what "all previous terms" means at each stage. Let's work through this systematically. The first term is 7, and the second term is 10. For the third term, you need the average of all previous terms (the first and second terms): $\frac{7 + 10}{2} = \frac{17}{2} = 8.5$ For the fourth term, you need the average of all three previous terms: $\frac{7 + 10 + 8.5}{3} = \frac{25.5}{3} = 8.5$ The fourth term is 8.5, which is answer choice B. Looking at the wrong answers: A) 8.0 might result from incorrectly averaging just the first and third terms, or from a calculation error. C) 8.75 could come from averaging only the second and third terms ( $\frac{10 + 8.5}{2} = 9.25$ ) and then making an error, or from other computational mistakes. D) 9.0 might result from averaging just the first two terms incorrectly or misunderstanding which terms to include. The key trap here is forgetting that "all previous terms" means you must include every term that came before, not just the most recent ones. Also, be careful with your arithmetic when dealing with decimals—double-check your addition and division. For sequence problems like this, always write out each term clearly and verify that you're including the correct previous terms in your calculations.
Which sequence follows the same rule as $5, 10, 15, 20, 25$ ?
Answer: $7, 12, 17, 22, 27$ This question tests SSAT Middle Level skills: identifying pattern rules from sequences. Patterns and sequences involve recognizing regular structures in numbers or shapes, often expressed as arithmetic or geometric progressions. In this sequence, numbers increase by 5 each time, and choice B follows the same arithmetic rule. Choice B is correct because it accurately matches the rule of adding 5, showing understanding of consistent arithmetic progressions. Choice A is incorrect because it uses multiplication by 2, reflecting a common misconception of applying geometric rules to arithmetic sequences. To help students: Encourage them to identify what changes between elements—look for operations or transformations. Practice with mixed sequences to see different types of patterns. Watch for: over-simplifying complex patterns or confusing similar operations.
Explain how the sequence is formed: $30, 27, 24, 21, 18$ .
Answer: Subtract 3 each time This question tests SSAT Middle Level skills: identifying pattern rules from sequences. Patterns and sequences involve recognizing regular structures in numbers or shapes, often expressed as arithmetic or geometric progressions. In this sequence, each number is decreased by 3 to produce the next. Choice A is correct because it accurately describes the sequence's rule, showing understanding of arithmetic progression with a negative common difference. Choice B is incorrect because it suggests adding 3, reflecting a common misconception of reversing the direction of change. To help students: Encourage them to identify what changes between elements—look for operations or transformations. Practice with mixed sequences to see different types of patterns. Watch for: over-simplifying complex patterns or confusing similar operations.
What is the rule for $2, 6, 18, 54, 162$ ?
Answer: Multiply by 3 each time This question tests SSAT Middle Level skills: identifying pattern rules from sequences. Patterns and sequences involve recognizing regular structures in numbers or shapes, often expressed as arithmetic or geometric progressions. In this sequence, each number is multiplied by 3 to produce the next. Choice C is correct because it accurately describes the sequence's rule, showing understanding of geometric progression with a common ratio of 3. Choice B is incorrect because it suggests multiplying by 2, reflecting a common misconception of underestimating the ratio. To help students: Encourage them to identify what changes between elements—look for operations or transformations. Practice with mixed sequences to see different types of patterns. Watch for: over-simplifying complex patterns or confusing similar operations.
Explain how the sequence is formed: $20, 18, 21, 19, 22, 20$ .
Answer: Subtract 2, then add 3 This question tests SSAT Middle Level skills: identifying pattern rules from sequences. Patterns and sequences involve recognizing regular structures in numbers or shapes, often expressed as arithmetic or geometric progressions. In this sequence, the pattern alternates between subtracting 2 and adding 3. Choice A is correct because it accurately describes the sequence's rule, showing understanding of alternating operations. Choice B is incorrect because it reverses the operations, reflecting a common misconception of misordering the pattern. To help students: Encourage them to identify what changes between elements—look for operations or transformations. Practice with mixed sequences to see different types of patterns. Watch for: over-simplifying complex patterns or confusing similar operations.
What is the rule for $6, 11, 16, 21, 26$ ?
Answer: Add 5 each time This question tests SSAT Middle Level skills: identifying pattern rules from sequences. Patterns and sequences involve recognizing regular structures in numbers or shapes, often expressed as arithmetic or geometric progressions. In this sequence, each number is increased by 5 to produce the next. Choice B is correct because it accurately describes the sequence's rule, showing understanding of arithmetic progression with a common difference of 5. Choice A is incorrect because it suggests adding 4, reflecting a common misconception of undercalculating the difference. To help students: Encourage them to identify what changes between elements—look for operations or transformations. Practice with mixed sequences to see different types of patterns. Watch for: over-simplifying complex patterns or confusing similar operations.
A sequence starts with 100 and each term is half the previous term. Which term will be the first term less than 1?
Answer: 8th term This problem involves a geometric sequence where each term is found by multiplying the previous term by a constant ratio. Here, you start with 100 and multiply by $\frac{1}{2}$ (or divide by 2) to get each subsequent term. Let's calculate each term systematically: - 1st term: 100 - 2nd term: $100 \times \frac{1}{2} = 50$ - 3rd term: $50 \times \frac{1}{2} = 25$ - 4th term: $25 \times \frac{1}{2} = 12.5$ - 5th term: $12.5 \times \frac{1}{2} = 6.25$ - 6th term: $6.25 \times \frac{1}{2} = 3.125$ - 7th term: $3.125 \times \frac{1}{2} = 1.5625$ - 8th term: $1.5625 \times \frac{1}{2} = 0.78125$ The 8th term (0.78125) is the first term less than 1, making C correct. Choice A suggests the 6th term (3.125), but this is still greater than 1. Choice B points to the 7th term (1.5625), which is also greater than 1, though getting close. Choice D indicates the 9th term, but you've already found a term less than 1 before reaching the 9th position. When working with geometric sequences that decrease rapidly like this one, write out each term methodically rather than trying to jump ahead. The pattern of halving creates smaller and smaller numbers, but it's easy to miscalculate mentally which specific term crosses your threshold.
What is the rule for $8, 12, 18, 27, 40, 60$ ?
Answer: Add 4, 6, 9, 13, 20 This question tests SSAT Middle Level skills: identifying pattern rules from sequences. Patterns and sequences involve recognizing regular structures in numbers or shapes, often expressed as arithmetic or geometric progressions. In this sequence, the differences are 4, 6, 9, 13, 20, which may follow an approximate multiplication by 1.5 with adjustments. Choice A is correct because it accurately lists the additions that generate the sequence, showing understanding of variable differences. Choice B is incorrect because it suggests multiplying by 2 each time, reflecting a common misconception of assuming a geometric progression. To help students: Encourage them to identify what changes between elements—look for operations or transformations. Practice with mixed sequences to see different types of patterns. Watch for: over-simplifying complex patterns or confusing similar operations.
The sequence 5, 8, 13, 20, 29, ... follows a specific pattern. Which expression represents the nth term of this sequence?
Answer: n² + 4 When you encounter a sequence problem, your goal is to identify the pattern by examining how the terms change and then test which algebraic expression generates those exact values. Let's analyze the sequence 5, 8, 13, 20, 29, ... by looking at the differences between consecutive terms: - 8 - 5 = 3 - 13 - 8 = 5 - 20 - 13 = 7 - 29 - 20 = 9 The differences form the sequence 3, 5, 7, 9, ... which increases by 2 each time. This pattern of increasing differences often indicates a quadratic relationship. Now let's test each expression by substituting n = 1, 2, 3, 4, 5: For choice A) $n^2 + 4$ : - n = 1: $1^2 + 4 = 5$ ✓ - n = 2: $2^2 + 4 = 8$ ✓ - n = 3: $3^2 + 4 = 13$ ✓ - n = 4: $4^2 + 4 = 20$ ✓ Choice B) $n^2 + 3n + 1$ gives 5, 11, 19, 29, ... (second term is wrong) Choice C) $2n^2 + 3$ gives 5, 11, 21, 35, ... (second term is wrong) Choice D) $n^2 + 2n + 2$ gives 5, 10, 17, 26, ... (second term is wrong) Only expression A produces the correct sequence values. Strategy tip: For sequence problems, always verify your answer by substituting the first few values of n. Don't just rely on pattern recognition—actually calculate to confirm the expression works for multiple terms.
The sequence 2, 6, 12, 20, 30, ... represents which pattern?
Answer: n² + n where n starts at 1 Test each rule by substituting $n = 1, 2, 3, \ldots$ and comparing with 2, 6, 12, 20, 30. For $n^2 + n$ : $1 + 1 = 2$ , $4 + 2 = 6$ , $9 + 3 = 12$ , $16 + 4 = 20$ , $25 + 5 = 30$ . Every term matches, so $n^2 + n$ is the pattern. (Each term is also the product of two consecutive whole numbers.) The rule $n^2 + 1$ gives 2, 5, 10, so it matches only the first term. The rule $2n^2 - 2n + 2$ is the trap: it gives 2 and 6, but then $18 - 6 + 2 = 14$ , not 12. The rule $2n(n+1)$ doubles every term (4, 12, 24), and $\frac{n(n+1)}{2}$ starting at $n = 2$ gives 3, 6, 10, which are triangular numbers. Always check at least three or four terms; two matching terms are not enough to confirm a pattern.
The differences between consecutive terms in a sequence are 2, 6, 18, 54, ... If the first term of the sequence is 5, what is the 5th term?
Answer: 85 When you encounter a sequence problem where you're given the differences between consecutive terms, you need to first identify the pattern in those differences, then use it to build the actual sequence. The differences given are 2, 6, 18, 54, ... Looking at these numbers, notice that each difference is 3 times the previous difference: $6 = 2 \times 3$ , $18 = 6 \times 3$ , $54 = 18 \times 3$ . So the next difference would be $54 \times 3 = 162$ . Now you can build the sequence starting with the first term of 5: - 1st term: 5 - 2nd term: $5 + 2 = 7$ - 3rd term: $7 + 6 = 13$ - 4th term: $13 + 18 = 31$ - 5th term: $31 + 54 = 85$ The answer is A) 85. Let's examine why the other choices are incorrect. B) 89 would result if you incorrectly added 4 extra to the correct answer, perhaps from a calculation error. C) 93 might come from mistakenly thinking the differences follow an arithmetic pattern (adding a constant each time) rather than the geometric pattern (multiplying by 3). D) 97 could result from various computational mistakes in building the sequence. Strategy tip: In sequence problems involving differences, always first identify the pattern in the differences themselves before building the actual sequence. Look for arithmetic patterns (constant addition) or geometric patterns (constant multiplication) in the difference sequence.
In the sequence 2, 5, 11, 23, 47, ..., each term follows the same pattern. What is the next term?
Answer: 95 When you encounter a sequence problem, your first step is to look for the pattern connecting consecutive terms. Don't just look at the differences between terms—sometimes the pattern involves the differences themselves. Let's examine the differences between consecutive terms: - 5 - 2 = 3 - 11 - 5 = 6 - 23 - 11 = 12 - 47 - 23 = 24 Now look at this sequence of differences: 3, 6, 12, 24... Each difference is doubling! This means the next difference should be 24 × 2 = 48. Therefore, the next term in the original sequence is 47 + 48 = 95. Let's check why the other answers don't work. Choice A (94) would require a difference of 47, which doesn't follow the doubling pattern of differences. Choice C (96) would need a difference of 49, and choice D (97) would need a difference of 50—neither fits the established pattern of doubling differences. The correct answer is B) 95. Here's your strategy for sequence problems: When the immediate differences between terms don't show an obvious pattern, examine the differences between those differences. Many SSAT sequence problems involve patterns in the "second level" of differences, such as constant increases, doubling (as here), or other multiplicative patterns. Always verify your pattern works for at least three consecutive steps before applying it to find the next term.
In a sequence, the first term is 3 and each subsequent term is found by adding 4 to the previous term, then subtracting 1. What is the 5th term of this sequence?
Answer: 15 When you encounter sequence problems, focus on identifying the pattern rule and applying it systematically to find each term. This sequence follows a two-step rule: add 4 to the previous term, then subtract 1. Let's work through each term carefully: Term 1: 3 (given) Term 2: 3 + 4 - 1 = 6 Term 3: 6 + 4 - 1 = 9 Term 4: 9 + 4 - 1 = 12 Term 5: 12 + 4 - 1 = 15 The 5th term is 15, making A correct. Looking at the wrong answers reveals common sequence mistakes. Choice B (18) likely comes from adding only 4 each time without subtracting 1, or from miscounting which term you're finding. Choice C (21) might result from adding 6 to each term (combining the +4 and -1 incorrectly as +6), or from calculation errors in the middle terms. Choice D (24) suggests adding 7 each time, possibly from misunderstanding the rule as "add 4, add 1" instead of "add 4, subtract 1." Notice that this sequence actually increases by 3 each time (since +4 - 1 = +3), so you could also solve it as an arithmetic sequence with first term 3 and common difference 3. However, always follow the rule as stated rather than looking for shortcuts, since sequence problems can have complex patterns that simple arithmetic progressions won't capture. For sequence questions, write out each term step-by-step using the given rule—this prevents careless errors and helps you spot patterns.
A sequence follows the pattern: 2, -4, 8, -16, 32, ... What is the 8th term?
Answer: -256 When you see a sequence with a clear pattern between consecutive terms, you're dealing with a geometric sequence. Look for what number you multiply by to get from one term to the next. Let's identify the pattern: From 2 to -4, you multiply by -2. From -4 to 8, you multiply by -2 again ( $-4 \times -2 = 8$ ). From 8 to -16, multiply by -2. This is a geometric sequence with first term $a_1 = 2$ and common ratio $r = -2$ . The formula for the nth term of a geometric sequence is $a_n = a_1 \times r^{(n-1)}$ . For the 8th term: $a_8 = 2 \times (-2)^7$ . Since $(-2)^7 = -128$ , we get $a_8 = 2 \times (-128) = -256$ . Let's check why the other answers are wrong. Choice A (-128) is what you'd get if you forgot to multiply by the first term—this is just $(-2)^7$ without the factor of 2. Choice C (256) gives you the right magnitude but the wrong sign; this happens if you miscalculate $(-2)^7$ as positive instead of negative. Choice D (512) occurs if you incorrectly think you need $(-2)^8$ instead of $(-2)^7$ —remember, the exponent is always one less than the term number. The correct answer is B (-256). Study tip: In geometric sequences, always double-check your signs when the common ratio is negative. Odd exponents give negative results, even exponents give positive results. Also remember the exponent in the formula is $(n-1)$ , not $n$ .
Predict the next number: $9, 18, 36, 72, \;?$
Answer: 144 This question tests SSAT Middle Level skills: identifying pattern rules from sequences. Patterns and sequences involve recognizing regular structures in numbers or shapes, often expressed as arithmetic or geometric progressions. In this sequence, each number is multiplied by 2 to produce the next. Choice C is correct because it accurately predicts the next number as 144, showing understanding of geometric progression with a common ratio of 2. Choice A is incorrect because it suggests 81, reflecting a common misconception of switching to a different ratio like 9 times something. To help students: Encourage them to identify what changes between elements—look for operations or transformations. Practice with mixed sequences to see different types of patterns. Watch for: over-simplifying complex patterns or confusing similar operations.
In an arithmetic sequence, the 4th term is 17 and the 8th term is 33. What is the 12th term?
Answer: 49 When you encounter arithmetic sequences, remember that they have a constant difference between consecutive terms. Your goal is to find this common difference and use it to locate any term in the sequence. Given that the 4th term is 17 and the 8th term is 33, you can find the common difference by recognizing that from the 4th to 8th term, there are 4 steps in the sequence. So: $\text{common difference} = \frac{33 - 17}{8 - 4} = \frac{16}{4} = 4$ Now you can find the 12th term. From the 8th term to the 12th term, there are 4 more steps: $\text{12th term} = 33 + 4(4) = 33 + 16 = 49$ Choice A (45) represents what you'd get if you miscalculated the common difference as 3 instead of 4, then added $33 + 3(4) = 45$ . Choice C (53) occurs if you incorrectly think there are 5 steps from the 8th to 12th term, calculating $33 + 4(5) = 53$ . Choice D (57) results from both errors combined: using a common difference of 3 and thinking there are 8 steps total, giving $33 + 3(8) = 57$ . The correct answer is B (49). Strategy tip: Always count the number of "jumps" carefully in sequence problems. From term 4 to term 8 is exactly 4 jumps, not 5. Drawing a simple diagram with the term positions can help you avoid this common counting error.
Identify the pattern in $2, 5, 10, 17, 26, 37$ .
Answer: Add 3, 5, 7, 9, 11 This question tests SSAT Middle Level skills: identifying pattern rules from sequences. Patterns and sequences involve recognizing regular structures in numbers or shapes, often expressed as arithmetic or geometric progressions. In this sequence, the differences increase by 2 each time, starting with adding 3, then 5, 7, 9, 11. Choice B is correct because it accurately describes the sequence's rule, showing understanding of increasing arithmetic differences. Choice A is incorrect because it suggests adding 3 constantly, reflecting a common misconception of ignoring the changing differences. To help students: Encourage them to identify what changes between elements—look for operations or transformations. Practice with mixed sequences to see different types of patterns. Watch for: over-simplifying complex patterns or confusing similar operations.
A sequence follows the pattern: 2, 6, 18, 54, ... Each term after the first is obtained by multiplying the previous term by the same number. What is the next term in the sequence?
Answer: 162 This is a geometric sequence problem, where each term is found by multiplying the previous term by a constant ratio. When you see a sequence with this pattern, your first step is to find the common ratio by dividing any term by the previous term. Let's find the common ratio: $6 ÷ 2 = 3$ , $18 ÷ 6 = 3$ , and $54 ÷ 18 = 3$ . The common ratio is 3, meaning each term is triple the previous term. To find the next term, multiply 54 by 3: $54 × 3 = 162$ . The correct answer is B. Now let's examine why the other choices are incorrect. Choice A (108) represents doubling the previous term instead of tripling it—this would be the pattern if the common ratio were 2, but we established it's 3. Choice C (216) would result from multiplying by 4, suggesting a misidentified common ratio. Choice D (270) would come from multiplying by 5, which is also an incorrect ratio. The key strategy for geometric sequences is always to verify your common ratio using multiple consecutive terms, not just one pair. If you had only checked $6 ÷ 2 = 3$ and missed checking the other ratios, you might have been less confident in your answer. Also, double-check your arithmetic—multiplying 54 by 3 carefully ensures you don't fall into a calculation trap. On the SSAT, sequence problems often include distractors that result from common computational errors or misidentified patterns.
What is the rule for $100, 90, 80, 70, 60$ ?
Answer: Subtract 10 each time This question tests SSAT Middle Level skills: identifying pattern rules from sequences. Patterns and sequences involve recognizing regular structures in numbers or shapes, often expressed as arithmetic or geometric progressions. In this sequence, each number is decreased by 10 to produce the next. Choice B is correct because it accurately describes the sequence's rule, showing understanding of arithmetic progression with a common difference of -10. Choice A is incorrect because it suggests subtracting 5, reflecting a common misconception of halving the actual difference. To help students: Encourage them to identify what changes between elements—look for operations or transformations. Practice with mixed sequences to see different types of patterns. Watch for: over-simplifying complex patterns or confusing similar operations.
What operation is used to generate $7, 14, 28, 56, 112$ ?
Answer: Multiply by 2 each time This question tests SSAT Middle Level skills: identifying pattern rules from sequences. Patterns and sequences involve recognizing regular structures in numbers or shapes, often expressed as arithmetic or geometric progressions. In this sequence, each number is multiplied by 2 to produce the next. Choice B is correct because it accurately describes the sequence's rule, showing understanding of geometric progression with a common ratio of 2. Choice A is incorrect because it suggests adding 7, reflecting a common misconception of mistaking multiplication for addition. To help students: Encourage them to identify what changes between elements—look for operations or transformations. Practice with mixed sequences to see different types of patterns. Watch for: over-simplifying complex patterns or confusing similar operations.

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