Predicting Later Terms - SHSAT Math

1. The first four terms of a sequence are 3, 7, 15, 31. If this pattern continues, which expression represents the nth term?
   Answer: $2^{n+1} - 1$ Examine the differences: 7-3=4, 15-7=8, 31-15=16. The differences are powers of 2: $2^2, 2^3, 2^4$. This suggests each term is related to powers of 2. Testing the pattern: 3 = $2^2 - 1 = 4 - 1$, 7 = $2^3 - 1 = 8 - 1$, 15 = $2^4 - 1 = 16 - 1$, 31 = $2^5 - 1 = 32 - 1$. So the nth term is $2^{n+1} - 1$. Choice B gives different values. Choice C is linear, not exponential. Choice D adds unnecessary complexity.
2. A piecewise sequence is defined as: $f(n) = \begin{cases} 2n + 1 & \text{if } n \text{ is odd} \\ n^2 - 1 & \text{if } n \text{ is even} \end{cases}$. What is the value of $f(15) + f(16)$?
   Answer: 286 Since 15 is odd: $f(15) = 2(15) + 1 = 30 + 1 = 31$. Since 16 is even: $f(16) = 16^2 - 1 = 256 - 1 = 255$. Therefore: $f(15) + f(16) = 31 + 255 = 286$. Choice B results from adding 1 extra. Choice C results from forgetting to subtract 1 from $16^2$. Choice D combines both errors.
3. The sequence $5,\;9,\;13,\;17,\;\dots$ follows a consistent pattern. What is the 50th term of the sequence?
   Answer: $201$ When you encounter a sequence like this, you're dealing with an arithmetic sequence—a pattern where the same number is added each time. Your first step is to identify the common difference by subtracting consecutive terms. Looking at the given sequence: $5, 9, 13, 17, \dots$ The common difference is $9 - 5 = 4$, and you can verify this: $13 - 9 = 4$ and $17 - 13 = 4$. So we're adding 4 each time. For any arithmetic sequence, the formula for the nth term is: $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference. Here, $a_1 = 5$ and $d = 4$, so: $a_{50} = 5 + (50-1) \times 4 = 5 + 49 \times 4 = 5 + 196 = 201$ Looking at the wrong answers: Choice A ($197$) comes from miscalculating $49 \times 4$ as $192$ instead of $196$. Choice C ($205$) results from using $n = 50$ instead of $(n-1) = 49$ in the formula, giving you $5 + 50 \times 4 = 205$. Choice D ($209$) compounds both errors: using $n$ instead of $(n-1)$ and then adding an extra 4. The correct answer is B: $201$. Study tip: Always double-check your arithmetic sequence formula—it's $(n-1)d$, not $nd$, because you don't add the common difference to get to the first term. Practice identifying the pattern first, then apply the formula carefully.
4. A Fibonacci-like sequence starts with $a_1 = 1$ and $a_2 = 3$, where each subsequent term equals the sum of the two preceding terms. If $a_n = 199$, what is the value of $a_{n+2}$?
   Answer: 521 First, generate the sequence: $a_1 = 1, a_2 = 3, a_3 = 4, a_4 = 7, a_5 = 11, a_6 = 18, a_7 = 29, a_8 = 47, a_9 = 76, a_{10} = 123, a_{11} = 199$. So $n = 11$. We need $a_{13}$. $a_{12} = a_{10} + a_{11} = 123 + 199 = 322$. $a_{13} = a_{11} + a_{12} = 199 + 322 = 521$. Choice B assumes an incorrect sequence position. Choice C uses wrong arithmetic. Choice D doubles one of the terms incorrectly.

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