Question

question
what is a formula for the nth term of the given sequence?
48, 72, 108...
answer
$a_n = 32(\frac{2}{3})^{1 - n}$ $a_n = 32(\frac{3}{2})^{-n}$
$a_n = 32(\frac{3}{2})^{n}$ $a_n = 48(\frac{2}{3})^{-n}$
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Explanation:

Step1: Identify the sequence type

This is a geometric sequence as $\frac{72}{48}=\frac{3}{2}$ and $\frac{108}{72}=\frac{3}{2}$, the common - ratio $r = \frac{3}{2}$. The first term $a_1=48$.

Step2: Recall the formula for the nth term of a geometric sequence

The formula for the nth term of a geometric sequence is $a_n=a_1r^{n - 1}$.

Step3: Substitute values into the formula

Substitute $a_1 = 48$ and $r=\frac{3}{2}$ into the formula: $a_n=48\times(\frac{3}{2})^{n - 1}$. We can rewrite $48$ as $32\times\frac{3}{2}$, so $a_n=32\times\frac{3}{2}\times(\frac{3}{2})^{n - 1}=32\times(\frac{3}{2})^{n}$.

Answer:

$a_n = 32(\frac{3}{2})^{n}$