Chapter 8, Problem 8Q

To determine

Find the rule for the $n\text{th}$ term of the sequence using graph.

Answer to Problem 8Q

The $n\text{th}$ term of the sequence is $a_n=0.5{\left(2\right)}^{n-1}$ .

Explanation of Solution

Given:

The given graph is

  

Calculation:

From the graph, the sequence is $0.5,1,2,4,8,---$

In a 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 $0.25,0.5,0.75,1,1.25,---$ .

  $\begin{array}{l}\frac{1}{0.5}=2\\ \frac{2}{1}=2\\ \frac{4}{2}=2\\ \frac{8}{4}=2\end{array}$

The common ratio $\left(2\right)$ is constant.

The general rule for the $n\text{th}$ term of the geometric sequence is $a_n=a_1{r}^{n-1}$ , where $a_1$ is first term and $r$ is the common ratio.

Substitute the value of $a_1=0.5$ and $r=2$ in the rule.

  $a_n=0.5{\left(2\right)}^{n-1}$

Hence the $n\text{th}$ term of the sequence is $a_n=0.5{\left(2\right)}^{n-1}$ .