Chapter 11.1, Problem 48PPS

To determine

Determine whether the sequence $-\frac{2}{5},-\frac{2}{25},-\frac{2}{125},-\frac{2}{625}$ is arithmetic, geometric, or neither.

Answer to Problem 48PPS

The sequence is geometric.

Explanation of Solution

Given:

The sequence:$-\frac{2}{5},-\frac{2}{25},-\frac{2}{125},-\frac{2}{625}$

Concept Used:

A sequence is a set of numbers, called terms, arranged in some particular order.

An arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference is called the common difference.

A geometric sequence is a sequence with the ratio between two consecutive terms constant.

Calculation:

First Term: $a_1=1$

Common Difference for Arithmetic sequence:

  $d=\text{2nd}\text{term}-\text{1st}\text{term =}-\frac{2}{25}-(-\frac{2}{5})=\frac{8}{25}\text{or}d=\text{3rd}\text{term}-\text{2nd}\text{term}\text{=}-\frac{2}{125}-(-\frac{2}{25})=\frac{8}{125}$

The common differences are not constant, so the sequence is not arithmetic.

Common Ratio for Geometric sequence: $r=\frac{\text{2nd}\text{term}}{\text{1st}\text{term}}=\frac{-\frac{2}{25}}{-\frac{2}{5}}=\frac{1}{5}\text{or}r=\frac{\text{3rd}\text{term}}{\text{2nd}\text{term}}=\frac{-\frac{2}{125}}{-\frac{2}{25}}=\frac{1}{5}$

The common ratios are constant, so the sequence is geometric.

Thus, the sequence is geometric.