Chapter 7, Problem 3T

Solution

To identify: Whether the sequence is arithmetic, geometric, or neither $40,10,\frac{5}{2},\frac{5}{8},\mathrm{...}$ .

The given sequence is a geometric sequence.

Given information:

The given sequence is $40,10,\frac{5}{2},\frac{5}{8},\mathrm{...}$ .

Explanation:

Consider the given sequence,

  $40,10,\frac{5}{2},\frac{5}{8},\mathrm{...}$

Firstly check if the given sequence is an arithmetic sequence or not. For doing so find the common difference of the consecutive term if the difference is constant then the sequence will be called an arithmetic sequence.

The common difference ( d ) is:

  $\begin{array}{c}10-40=-30\\ \frac{5}{2}-10=-\frac{15}{2}\end{array}$

The given sequence is not an arithmetic sequence because the common difference of two successive terms is not constant.

Now check if the given sequence is a geometric sequence or not. For doing so find the common ratio of the consecutive term if the ratio is constant then the sequence will be called a geometric sequence.

The common ratio ( r ) is:

  $\begin{array}{l}r=\frac{10}{40},r=\frac{1}{4}\\ r=\frac{\frac{5}{2}}{10},r=\frac{1}{4}\\ r=\frac{\frac{5}{8}}{\frac{5}{2}},r=\frac{10}{40},r=\frac{1}{4}\end{array}$

The given sequence is geometric sequence because the common ratio of two successive terms is a constant.

Therefore, $40,10,\frac{5}{2},\frac{5}{8},\mathrm{...}$ is a geometric sequence.