Chapter 11.5, Problem 10WE

To determine

To calculate:

Sum of given arithmetic series.

Answer to Problem 10WE

Sum of given arithmetic series is $300$ .

Explanation of Solution

Give information:

  $100+98+96+94+\mathrm{...}$

Calculation:

Given series is,

  $100+98+96+94+\mathrm{...}$

Her, $a_1=100$ and common difference is $d=98-100=-2$ last term given is $94$ ,

The $n^{th}$ term is given by,

  $a_n=a_1+\left(n-1\right)d$

Put the values in above equation to get ${100}^{th}$ term,

  $\begin{array}{l}100=100+\left(n-1\right)-2\\ 100-100=n-1-2\\ 0=n-3\\ n=3\end{array}$

The sum of series is given by,

  $\begin{array}{l}=\frac{n}{2}\left(a_1+a_n\right)\\ =\frac{3}{2}\left(100+100\right)\\ =1.5\left(200\right)\\ =300\end{array}$

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