Chapter 12.2, Problem 16E

To determine

To calculate:

The common ratio and next three term of the geometric sequence $10,2,0.4,\mathrm{...}$

Answer to Problem 16E

The common ratio is $\frac{1}{5}$ and next three terms of the geometric sequence are $0.08,0.016,0.0032$

Explanation of Solution

Given information:

  $10,2,0.4,\mathrm{...}$

Formula used:

The common ratio is given by the following expression:

  $r=\frac{a_n}{a_{n-1}}$

The $n^{th}$ term of the geometric sequence is given by the following expression:

  $a_n=a_1{r}^{n-1}$

Calculation:

The given geometric sequence is considered as:

  $\begin{array}{l}{a}_1=10\\ a_2=2\\ a_3=0.4\end{array}$

Now, the common ratio is calculated as:

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{2}{10}=\frac{1}{5}\\ r=\frac{a_3}{a_2}=\frac{0.4}{2}=\frac{1}{5}\end{array}$

The next three terms of the geometric sequence are calculated as:

  $\begin{array}{l}{a}_4=a_1{r}^{4-1}=10{\left(\frac{1}{5}\right)}^3=0.08\\ a_5=a_1{r}^{5-1}=10{\left(\frac{1}{5}\right)}^4=0.016\\ a_6=a_1{r}^{6-1}=10{\left(\frac{1}{5}\right)}^5=0.0032\end{array}$