Question

Find the sum of all natural numbers lying between $100$ and $1000$, which are multiples of $5$.

Answer 1

First and last natural number lying between $100$ and $1000$ (excluding these two) are $105$ and $995$ respectively
$\Rightarrow a=105,d=5,a+(n-1)d=995$
$\Rightarrow 105+(n-1)5=995\Rightarrow (n-1)5=995-105=890\Rightarrow n-1=178\Rightarrow n=179$
$\therefore S_n=\frac{179}{2}\left[2\right(105)+(179-1\left)\right(5\left)\right]=\frac{179}{2}\left[2\right(105)+(178\left)\right(5\left)\right]=179[105+(89\left)5\right]=\left(179\right)(105+445)=\left(179\right)\left(550\right)=98450$
Thus the sum of all natural numbers lying between $100$ and $1000$, which are multiples of $5$, is $98450$.