Chapter 8, Problem 25CR

To determine

Find the first six terms of the given sequence.

Answer to Problem 25CR

The first six terms of the sequence are

  $9,6,4,\frac{8}{3},\frac{16}{9},\frac{32}{27}$

Explanation of Solution

Given:

The given sequence is $9,6,4,\frac{8}{3},\frac{16}{9},---$ .

Calculation:

In an geometric sequence, the ratio of any term to the previous term, called the common ratio, is constant.

Find the common ratio of any term to the previous term of the sequence $9,6,4,\frac{8}{3},\frac{16}{9},---$ .

  $\begin{array}{l}\frac{6}{9}=\frac{2}{3}\\ \frac{4}{6}=\frac{2}{3}\\ \frac{\frac{8}{3}}{4}=\frac{2}{3}\\ \frac{\frac{16}{9}}{\frac{8}{3}}=\frac{2}{3}\end{array}$

The common ratio $\left(\frac{2}{3}\right)$ is constant.

  $nth$ term of the series is $a_n=a_1{r}^{n-1}$ .

  $\begin{array}{l}{a}_6=9{\left(\frac{2}{3}\right)}^{6-1}\\ =9{\left(\frac{2}{3}\right)}^5\\ =9\cdot \frac{32}{243}\\ =\frac{32}{27}\end{array}$

Hence the first six terms of the sequence are

  $9,6,4,\frac{8}{3},\frac{16}{9},\frac{32}{27}$