Chapter 9, Problem 46RE

Solution

To calculate: The sum of the arithmetic series $-5,-3,-1,1,\mathrm{.....},55$ .

The sum of the series $2.5,-0.5,-3.5,\mathrm{.....},-75.5=-985.5$ .

Given information:

The given terms are $-5,-3,-1,1,\mathrm{.....},55$ .

Formula used:

The sum nth term of an arithmetic sequence with first term $a_1$ and $a_7$ is given by: $s_n=n\left(\frac{a_1+a_n}{2}\right)$ .

Calculation:

Consider the given sequence.

It is given that $-5,-3,-1,1,\mathrm{.....},55$ .

Determine the value of common difference.

  $\begin{array}{l}d=a_2-a_1\\ d=-3-\left(-5\right)\\ d=2\end{array}$

Therefore $a_1=-5$ and $d=2$ .

Determine the $a_n$ by using $nth$ term formula.

  $\begin{array}{c}{a}_n=a+\left(n-1\right)d\\ 55=-5\left(n-1\right)\left(2\right)\\ 55=5+2n-2\\ 55=2n-7\\ 2n=62\\ n=\frac{62}{2}\\ n=31\end{array}$

The given sequence is arithmetic with first term $a_1=-5\text{and}{a}_{31}=55$ .

Put the value of $a_1=-5,a_{31}=55\text{and}d=2$ into arithmetic sum formula.

  $\begin{array}{c}{s}_{31}=n\left(\frac{a_1+a_{31}}{2}\right)\\ s_{31}=\frac{31\left(-5+55\right)}{2}\\ s_{31}=\frac{31}{2}\cdot \left(50\right)\\ s_{31}=775\end{array}$

So, the sum of the series is $775$ .