Chapter 12, Problem 22RE

To determine

To find: Whether the given sequence is arithmetic, geometric or neither. Determine the common difference and the sum of first n terms if the sequence is arithmetic. Determine the common ratio and the sum of First n terms if the sequence is geometric.

Answer to Problem 22RE

The sequence is geometric with common ration of $-3$ and the sum of first n term is $\frac{5}{4}\left[1-{\left(-3\right)}^n\right]$ .

Explanation of Solution

Given:

The given series is $5,-\frac{5}{3},\frac{5}{9},-\frac{5}{27},\frac{5}{81},\mathrm{....}$ .

Calculation:

Consider the given series is,

  $5,-\frac{5}{3},\frac{5}{9},-\frac{5}{27},\frac{5}{81},\mathrm{....}$

The difference of the first and the second term is,

  $\frac{-5}{3}-5=\frac{-20}{3}$

The difference of the third and the second term is,

  $\frac{5}{9}+\frac{5}{3}=\frac{15}{9}$

Thus, the sequence is not arithmetic.

The common ratio of the first and the second term is,

  $\frac{\frac{-5}{3}}{5}=-3$

The common ratio of the second and the third term is,

  $\frac{\frac{5}{9}}{\frac{-5}{3}}=-3$

Thus, the sequence is geometric.

The sum of the first n term of the geometric sequence is,

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

Then,

  $\begin{array}{c}{S}_n=5\left(\frac{1-{\left(-3\right)}^n}{1-\left(-3\right)}\right)\\ =\frac{5}{4}\left[1-{\left(-3\right)}^n\right]\end{array}$