Chapter EP, Problem 11.3.9EP

To determine

To write: The equation for the $n^{\text{th}}$ term of the given geometric sequence.

Answer to Problem 11.3.9EP

The equation for the $n^{\text{th}}$ term of the geometric sequence is $a_n=-\frac{1}{2}{\left(\frac{1}{4}\right)}^{n-1}$ .

Explanation of Solution

Given: The given sequence is $-\frac{1}{2},-\frac{1}{8},-\frac{1}{32}$ .

The $n^{th}$ term of a geometric sequence is given by,

$a_n=a_1{r}^{n-1}$ ...... (1)

The first term of the sequence is,

$a_1=-\frac{1}{2}$

The common ratio is calculated as,

$\begin{array}{c}r=\frac{-1/8}{-1/2}\\ =\frac{2}{8}\\ =\frac{1}{4}\end{array}$

Substitute the values equation (1).

$a_n=-\frac{1}{2}{\left(\frac{1}{4}\right)}^{n-1}$

Therefore, the equation for the $n^{\text{th}}$ term of the geometric sequence is $a_n=-\frac{1}{2}{\left(\frac{1}{4}\right)}^{n-1}$ .