Question

Find the sum of all the numbers between $100$ and $200$ which are divisible by $7$.

Answer 1

The numbers which are divisible by $7$ between $100$ and $200$ are $105,112,119,126,...,196$

The given series is in A.P. Since, the difference between the consecutive terms is constant. i.e., $11-105=119-112=...=7$

The first term $a=105$

The common difference $d=7$

The last term $a_n=196$

$⟹a+(n-1)d=196$

$⟹105+(n-1)7=196$

$⟹(n-1)7=91$

$⟹n-1=13$

$⟹n=14$

Therefore the sum the terms is $S_n=\frac{n}{2}(a+a_n)$

$⟹S_n=\frac{14}{2}(105+196)$

$⟹S_n=7\left(301\right)=2107$

Therefore, the sum of all the numbers between $100$ and $200$ which are divisible by $7$ is $2107$