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Advanced Math Difficulty: Hard

The first term of a sequence is $9$ . Each term after the first is $4$ times the preceding term. If $w$ represents the $n$ th term of the sequence, which equation gives $w$ in terms of $n$ ?

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Explanation

Choice D is correct. Since $w$ represents the $n$ th term of the sequence and $9$ is the first term of the sequence, the value of $w$ is $9$ when the value of $n$ is $1$ . Since each term after the first is $4$ times the preceding term, the value of $w$ is $9 left parenthesis 4 right parenthesis$ when the value of $n$ is $2$ . Therefore, the value of $w$ is $9 left parenthesis 4 right parenthesis left parenthesis 4 right parenthesis$ , or $9 left parenthesis 4 right parenthesis squared$ , when the value of $n$ is $3$ . More generally, the value of $w$ is $9 left parenthesis 4 Superscript n minus 1 Baseline right parenthesis$ for a given value of $n$ . Therefore, the equation $w = 9 (4^{n - 1})$ gives $w$ in terms of $n$ .

Choice A is incorrect. This equation describes a sequence for which the first term is $36$ , rather than $9$ , and each term after the first is $9$ , rather than $4$ , times the preceding term.

Choice B is incorrect. This equation describes a sequence for which the first term is $4$ , rather than $9$ , and each term after the first is $9$ , rather than $4$ , times the preceding term.

Choice C is incorrect. This equation describes a sequence for which the first term is $36$ , rather than $9$ .