Question

Find the sum of odd integers from $1$ to $2001$.

Answer 1

The odd integers from$1$ to $2001$ are $1,3,5,...1999,2001$.

This sequence form an A.P.

Here, first term is, $a=1$ and common difference, $d=2$

Here $a+(n-1)d=2001$

$\Rightarrow 1+(n-1)\left(2\right)=2001$

$\Rightarrow 2n-2=2000$

$\Rightarrow n=1001$

Hence required sum is,

$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

$=\frac{1001}{2}[2\times 1+(1001-1)\times 2]$

$=\frac{1001}{2}[2+1000\times 2]$

$=\frac{1001}{2}\times 2002$

$=1001\times 1001$
$=1002001$