Question

Find the sum of even positive integers between 1 and 200.

Answer 1

Let , The sum of even positive integers between $1$ and $200$ be $S\left(n\right)$

The given are the even numbers between $1$ and $200$ in the form of AP :

$2,4,6.......198$

Here,

First term, $a=2$

Common Difference, $d=a\left(2\right)-a\left(1\right)=(4-2)=2$

Last term, $L=198$

We know,

$L=a+(n-1)d$

$198=2+(n-1)2$

$198=2+2n-2$

$198=2n$

$n=\frac{198}{2}$

$n=99$

We know,

$S\left(n\right)=\frac{n}{2}(a+L)$

$S\left(99\right)=\frac{99}{2}(2+198)$

$=99\times 100$

$=9900$

Thus, the sum of even positive integers between $1$ and $200$ is $9900$