Chapter 11.5, Problem 8WE

To determine

To calculate:

Sum of given arithmetic series.

Answer to Problem 8WE

Sum of given arithmetic series is $300$ .

Explanation of Solution

Give information:

  ${\displaystyle \sum _{m=10}^{20}\left(30-m\right)}$

Calculation:

Given series is,

  $={\displaystyle \sum _{m=10}^{20}\left(30-m\right)}$

To find out sum of given sequence,

  $S_n=\left(\frac{a_1+a_n}{2}\right)n$

Where,

  $S_n=$ sum of $n$ terms,

Here,

  $\begin{array}{l}n=20\\ a_1=\left(30-m\right)=\left(30-10\right)=20\\ a_n=\left(30-m\right)=\left(30-20\right)=10\end{array}$

Put the values,

  $\begin{array}{l}{S}_n=\left(\frac{a_1+a_n}{2}\right)n\\ S_{20}=\left(\frac{20+10}{2}\right)20\\ S_{20}=\left(\frac{30}{2}\right)20\\ S_{20}=\left(15\right)20\\ S_{20}=300\end{array}$

Therefore, Sum of given arithmetic series is $300$ .