Chapter 8, Problem 7CT

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 7CT

The given sequence is neither arithmetic nor geometric.

Explanation of Solution

Given:

The given graph is

  

Calculation:

From the graph, the sequence is $\frac{1}{4},\frac{5}{12},\frac{1}{2},\frac{13}{24}---$ .

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

Find the common difference between terms in the sequence $\frac{1}{4},\frac{5}{12},\frac{1}{2},\frac{13}{24}---$ .

  $\begin{array}{l}\frac{5}{12}-\frac{1}{4}=\frac{5-3}{12}=\frac{2}{12}=\frac{1}{6}\\ \frac{1}{2}-\frac{5}{12}=\frac{6-5}{12}=\frac{1}{12}\end{array}$

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 $\frac{1}{4},\frac{5}{12},\frac{1}{2},\frac{13}{24}---$ .

  $\begin{array}{l}\frac{\frac{5}{12}}{\frac{1}{4}}=\frac{5}{3}\\ \frac{\frac{1}{2}}{\frac{5}{12}}=\frac{6}{5}\end{array}$

The common ratio is not constant.

Hence the given sequence is neither arithmetic nor geometric.