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

Given:

Odd integers between 1 and 1000

To do:

We have to show the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.

Solution:

Odd integers between 1 and 1000 which are divisible by 3 are \( 3,9,15 \ldots, 999 \).

The sequence is in A.P.

Here,

\( a=3 \) and \( d=9-3=6 \) \( l=999 \)

We know that,

$l=a+(n-1) d$

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

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

$\Rightarrow 999+3=6 n$

$\Rightarrow n=\frac{1002}{6}=167$

$\therefore n=167$

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

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

$=\frac{167}{2}[6+166 \times 6]$

$=\frac{167}{2}(1002)$

$=167 \times 501$

$=83667$

The sum of all odd integers between 1 and 1000 which are divisible by 3 is $83667$.