Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.

The odd integers between 1 and 1000 that are divisible by 3 are:

3, 9, 15, 21...999

Here, we have:

$a=3, d=6$

$a_{n}=999$

$\Rightarrow 3+(n-1) 6=999$

$\Rightarrow 3+6 n-6=999$

$\Rightarrow 6 n=1002$

$\Rightarrow n=167$

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

$\Rightarrow S_{167}=\frac{167}{2}[2 \times 3+(167-1) 6]$

$\Rightarrow S_{167}=\frac{167}{2}[1002]=83667$

Hence proved.