Question

Find the sum of integers from $11$ to $1000$ which are divisible by $3$

Answer 1

Integers that are divisible by $3$ from $11$ to $1000$ are
$12,15,\mathrm{18............................................999}$
Now, Sum $=12+15+\mathrm{18...................................}+999$
$=3(4+5+\mathrm{6...................................}+333)$
We know that, sum of first $n$ integers $=\frac{n(n+1)}{2}$
It can also be written as
$S=3\left[\right(1+2+3+4+5+\mathrm{6...................................}+333)-(1+2+3\left)\right]$
$=3[(\frac{333\left(334\right)}{2})-\left(6\right)]$
$=3\times \left[55605\right]$
$=166815$