Question

Find the sum of all natural numbers between 200 and 400 which are divisible by 7. [CBSE 2012]

Answer 1

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,

$$\begin{gathered} a_n=399 \\ \Rightarrow 203+\left(n-1\right)\times 7=399\left[a_n=a+\left(n-1\right)d\right] \\ \Rightarrow 7n+196=399 \\ \Rightarrow 7n=399-196=203 \end{gathered}$$
$\Rightarrow n=29$

∴ Required sum = $\frac{29}{2}\left(203+399\right)\left[S_n=\frac{n}{2}\left(a+l\right)\right]$
$$\begin{gathered} =\frac{29}{2}\times 602 \\ =8729 \end{gathered}$$

Hence, the required sum is 8729.