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

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

This sequence forms an $A.P.$

Here, first term, $a=1$

Common difference, $d=2$

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

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

$\Rightarrow 2 n-2=2000$

$\Rightarrow n=1001$

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

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

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

$=1001 \times 1001$

$=1002001$

Thus, the sum of odd numbers from $1$ to $2001$ is $1002001 .$

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