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

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The numbers lying between $200$ and $400 ,$ which are divisible by $7,$ are $203,210,217 \ldots .399$

$\therefore$ First term, $a=203$

Last term, $I=399$

Common difference, $d=7$

Let the number of terms of the $A.P.$ be $n.$

$\therefore a_{n}=399=a+(n-1) d$

$\Rightarrow 399=203+(n-1) 7$

$\Rightarrow 7(n-1)=196$

$\Rightarrow n-1=28$

$\Rightarrow n=29$

$\therefore S_{29}=\frac{29}{2}(203+399)$

$=\frac{29}{2}(602)$

$=(29)(301)$

$=8729$

Thus, the required sum is $8729 .$

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