Chapter 8, Problem 6CT

To determine

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

Answer to Problem 6CT

The given sequence is arithmetic and the $n\text{th}$ term of the sequence is $a_n=11+\left(n-1\right)\left(-2\right)$ .

Explanation of Solution

Given:

The given graph is

  

Calculation:

From the graph, the sequence is $11,9,7,5---$ .

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

Find the common difference between terms in the sequence $11,9,7,5---$ .

  $\begin{array}{l}9-11=-2\\ 7-9=-2\\ 5-7=-2\end{array}$

The common difference $\left(-2\right)$ isconstant.

The general rule for the $n\text{th}$ term of the arithmetic sequence is $a_n=a_1+\left(n-1\right)d$ , where $a_1$ is first term and $d$ is the common difference.

Substitute the value of $a_1=1$ and $d=-2$ in the rule.

  $a_n=11+\left(n-1\right)\left(-2\right)$

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 $11,9,7,5---$ .

  $\begin{array}{l}\frac{9}{11}=\frac{9}{11}\\ \frac{7}{9}=\frac{7}{9}\end{array}$

The common ratio is constant.

Hence the given sequence is arithmetic and the $n\text{th}$ term of the sequence is $a_n=11+\left(n-1\right)\left(-2\right)$ .