Chapter 12, Problem 1RE

To determine

To find: The first five terms of each sequence.

Answer to Problem 1RE

The first five terms are $-\left(\frac{4}{3}\right)$ , $\left(\frac{5}{4}\right)$ , $-\left(\frac{6}{5}\right)$ , $\left(\frac{7}{6}\right)$ and $-\left(\frac{8}{7}\right)$ .

Explanation of Solution

Given:

The termis $\left\{a_n\right\}=\left\{{\left(-1\right)}^n\left(\frac{n+3}{n+2}\right)\right\}$ .

Calculation:

The first five terms are calculated as,

  $\begin{array}{c}\left\{a_1\right\}=\left\{{\left(-1\right)}^1\left(\frac{1+3}{1+2}\right)\right\}\\ =-\left(\frac{4}{3}\right)\end{array}$

  $\begin{array}{c}\left\{a_2\right\}=\left\{{\left(-1\right)}^2\left(\frac{2+3}{2+2}\right)\right\}\\ =\left(\frac{5}{4}\right)\end{array}$

  $\begin{array}{c}\left\{a_3\right\}=\left\{{\left(-1\right)}^3\left(\frac{3+3}{3+2}\right)\right\}\\ =-\left(\frac{6}{5}\right)\end{array}$

  $\begin{array}{c}\left\{a_4\right\}=\left\{{\left(-1\right)}^4\left(\frac{4+3}{4+2}\right)\right\}\\ =\left(\frac{7}{6}\right)\end{array}$

  $\begin{array}{c}\left\{a_5\right\}=\left\{{\left(-1\right)}^5\left(\frac{5+3}{5+2}\right)\right\}\\ =-\left(\frac{8}{7}\right)\end{array}$

Therefore, the first five terms are $-\left(\frac{4}{3}\right)$ , $\left(\frac{5}{4}\right)$ , $-\left(\frac{6}{5}\right)$ , $\left(\frac{7}{6}\right)$ and $-\left(\frac{8}{7}\right)$ .