Chapter 11.5, Problem 5WE

To determine

To calculate:

Sum of given arithmetic series.

Answer to Problem 5WE

Sum of given arithmetic series is $25250$ .

Explanation of Solution

Give information:

  ${\displaystyle \sum _{k=1}^{100}5k}$

Calculation:

Given series is,

  $\begin{array}{l}={\displaystyle \sum _{k=1}^{100}5k}\\ ={\displaystyle \sum _{k=1}^{100}5\left(1\right)}\\ ={\displaystyle \sum _{k=1}^{100}5}\end{array}$

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,

  $\begin{array}{l}{a}_1=5k=5\left(1\right)=5\\ a_n=5k=5\left(100\right)=500\end{array}$

Put the values,

  $\begin{array}{l}{S}_n=\left(\frac{a_1+a_n}{2}\right)n\\ S_n=\left(\frac{5+500}{2}\right)100\\ S_n=\left(\frac{505}{2}\right)100\\ S_n=\left(252.5\right)100\\ S_n=25250\end{array}$

Therefore, Sum of given arithmetic series is $25250$ .