Chapter 8, Problem 15CR

To determine

Find the $n\text{th}$ term of the given geometric sequence and graph the first six terms.

Answer to Problem 15CR

The $n\text{th}$ term $a_n=8-\left(n-1\right)4$

Explanation of Solution

Given:

The given sequenceis $25,10,4,\frac{8}{5},---$ .

Calculation:

The general rule for the $n\text{th}$ term is $a_n=a_1{r}^{n-1}$ .

Where $a_1\to$ first term, $r\to$ common ratio, $n\to$ number of terms.

In the given sequence $25,10,4,\frac{8}{5},---$

  $\begin{array}{l}{a}_1=15\\ r=\frac{10}{25}=\frac{2}{5}\end{array}$

The $n\text{th}$ term

  $a_n=25{\left(\frac{2}{5}\right)}^{n-1}$

The first $\text{6}$ terms are

  $\begin{array}{l}{a}_1=25\\ a_2=25\cdot \frac{2}{5}=10\\ a_3=10\cdot \frac{2}{5}=4\\ a_4=4\cdot \frac{2}{5}=\frac{8}{5}\\ a_5=\frac{8}{5}\cdot \frac{2}{5}=\frac{16}{25}\\ a_6=\frac{16}{25}\cdot \frac{2}{5}=\frac{32}{125}\end{array}$

Graph of the first six terms is given below.