Find the sum of all natural numbers between 200 and 400 which are divisible by 7.
Question:

Find the sum of all natural numbers between 200 and 400 which are divisible by 7.

Solution:

Natural numbers between 200 and 400 which are divisible by 7 are 203, 210, …, 399.

This is an AP with a = 203, d = 7 and l = 399.

Suppose there are n terms in the AP. Then,

$a_{n}=399$

$\Rightarrow 203+(n-1) \times 7=399 \quad\left[a_{n}=a+(n-1) d\right]$

$\Rightarrow 7 n+196=399$

$\Rightarrow 7 n=399-196=203$

$\Rightarrow n=29$

$\therefore$ Required sum $=\frac{29}{2}(203+399) \quad\left[S_{n}=\frac{n}{2}(a+l)\right]$

$=\frac{29}{2} \times 602$

$=8729$

Hence, the required sum is 8729.