Find the sum of all integers between 50 and 500, which are divisible by 7.

Given:

Integers between 50 and 500, which are divisible by 7.

To do:

We have to find the sum of all integers between 50 and 500, which are divisible by 7.

Solution:

Integers between 50 and 500, which are divisible by 7 are \( 56,63,70, \ldots, 497 \).

The sequence is in A.P.

Here,

\( a=56 \) and \( d=63-56=7 \) \( l=497 \)

We know that,

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

$\Rightarrow 497=56+(n-1) \times 7$

$\Rightarrow 497=56+7n-7$

$\Rightarrow 497-49=7 n$

$\Rightarrow n=\frac{448}{7}=64$

$\therefore n=64$

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

$=\frac{64}{2}[2 \times 56+(64-1) \times 7]$

$=32[112+63 \times 7]$

$=32(112+441)$

$=32 \times 553$

$=17696$

The sum of all integers between 50 and 500 which are divisible by 7 is $17696$.