Chapter 11.2, Problem 39PPS

To determine

Find the sum of the first 100 even natural numbers

Answer to Problem 39PPS

The sum first 100 even terms is 10100.

Explanation of Solution

Given:

Find the sum of the first 100 even natural numbers Concept Used:

The general formula for sum of arithmetic series is: ${\text{S}}_{\text{n}}=\frac{n}{2}[2a_1+(n-1)·d]$

If nth term is given, then the formula: ${\text{S}}_{\text{n}}=\frac{n}{2}(a_1+a_n)$

Calculation:

The series: $2,4,6,\mathrm{......100}\text{even}\text{terms}$

  $a_1=2\text{and}d=4-2=2,n=100;\text{find}{\text{S}}_{\text{n}}$

  $\begin{array}{l}{a}_1=2\text{and}d=2,n=100;\text{find}{\text{S}}_{\text{n}}\\ {\text{S}}_{\text{n}}=\frac{n}{2}[2a_1+(n-1)·d]\\ {\text{S}}_{\text{n}}=\frac{100}{2}[2·2+(100-1)·2]\\ {\text{S}}_{\text{n}}=50[4+198]=50·202=10100\\ {\text{S}}_{\text{n}}=10100\end{array}$

Thus, the sum first 100 even terms is 10100.