Chapter 11, Problem 23SGR

To determine

Find the sum of the series: $16+7+-2+\mathrm{.........}+-65$

Answer to Problem 23SGR

The sum of the series is $-245$ .

Explanation of Solution

Given:

The series: $16+7+-2+\mathrm{.........}+-65$

Concept Used:

The nth term of AP series: $a_n=a_1+(n-1)·d$

The sum of nth term of AP series: ${\text{S}}_n=\frac{n}{2}[a_1+a_n]$

Calculation:

  $\begin{array}{l}{a}_1=16;d=7-16=-9;a_n=-65;\text{find}n\\ a_n=a_1+(n-1)·d\\ -65=16+(n-1)·-9\\ -65=16-9n+9\\ -65=25-9n\\ -65-25=-9n+25-25\\ -90=-9n\\ \frac{-90}{-9}=\frac{-9n}{-9}\\ n=10\end{array}$

  $\begin{array}{l}{\text{S}}_n=\frac{n}{2}[a_1+a_n]\\ a_1=16;a_n=-65;n=10;\text{find}{\text{S}}_n\\ {\text{S}}_n=\frac{n}{2}[a_1+a_n]\\ {\text{S}}_n=\frac{10}{2}[16-65)]\\ {\text{S}}_n=5·-49=-245\\ {\text{S}}_n=-245\end{array}$

Thus, the sum of the series is $-245$ .