Question

In Problems 31–38, find the nth term $a_n$of each geometric sequence. When given, r is the common ratio.

$4,1,\frac{1}{4},\frac{1}{16},...$

Short answer

The common ratio is$r=\frac{1}{4}$and the nth term is$4·{\frac{1}{4}}^{n-1}$

Step-by-step solution

Step 1. Write the given information.

The geometric sequence is$4,1,\frac{1}{4},\frac{1}{16},...$

Step 2. Find the common ratio.

$$\begin{gathered} a_1=4,a_2=1,a_3=\frac{1}{4} \\ \frac{a_2}{a_1}=\frac{1}{4} \\ \frac{a_3}{a_2}=\frac{{\displaystyle \frac{1}{4}}}{1} \\ =\frac{1}{4} \\ r=\frac{1}{4} \end{gathered}$$

Step 3. Find the nth term.

The formula for the nth term is $a_n=a_1·r^{n-1}$

$a_n=4·{\frac{1}{4}}^{n-1}$