Chapter 8, Problem 5CT

To determine

Check whether the given graph represent an arithmetic sequence, geometric sequence, or neither. Find the rule for the $n\text{th}$ term of the sequence.

Answer to Problem 5CT

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

Explanation of Solution

Given:

The given graph is

  

Calculation:

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

In an arithmetic sequence, the difference of consecutive terms, called the common difference, is constant.

Find the common difference between terms in the sequence $1,2,4,8---$ .

  $\begin{array}{l}2-1=1\\ 4-2=2\end{array}$

The common difference is not constant.

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 $1,2,4,8---$ .

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

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 $a_1=1$ and $r=2$ in the rule.

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

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