Chapter 12, Problem 10CT

To determine

Whether the sequence $25,10,4,\frac{8}{5},\mathrm{...}$ is arithmetic, geometric, or neither. If the sequence is arithmetic find the common difference and the sum of the first n terms. If the sequence is geometric find the common ratio and sum of the first n terms.

Answer to Problem 10CT

Solution:

The sequence $25,10,4,\frac{8}{5},\mathrm{...}$ is geometric with common ratio $r=\frac{2}{5}$ and the sum of first n terms is $\frac{125}{3}\cdot \left(1-{\left(\frac{2}{5}\right)}^n\right)$.

Explanation of Solution

Given information:

Sequence, $25,10,4,\frac{8}{5},\mathrm{...}$

Explanation:

Consider the sequence $25,10,4,\frac{8}{5},\mathrm{...}$

Calculate the ratio of consecutive terms:

$\frac{10}{25}=\frac{2}{5},\frac{4}{10}=\frac{2}{5},\frac{\frac{8}{5}}{4}=\frac{2}{5}$

The ratio of consecutive terms is constant. Hence the sequence is geometric with common ratio

$r=\frac{2}{5}$ and first term $a_1=25$.

The sum of the first n terms of the sequence is given by:

$S_n=a_1\cdot \frac{1-r^n}{1-r}=25\cdot \frac{\left(1-{\left(\frac{2}{5}\right)}^n\right)}{1-\frac{2}{5}}$

$=25\cdot \frac{\left(1-{\left(\frac{2}{5}\right)}^n\right)}{\frac{3}{5}}$

$=\frac{125}{3}\cdot \left(1-{\left(\frac{2}{5}\right)}^n\right)$

Thus, the sequence $25,10,4,\frac{8}{5},\mathrm{...}$ is geometric with common ratio $r=\frac{2}{5}$ and the sum of first n terms is $\frac{125}{3}\cdot \left(1-{\left(\frac{2}{5}\right)}^n\right)$.