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# In an arithmetic progression, if the $k^{th}$ term is $5k+1$, then find the sum of first $100$ terms.

**Given:**In an arithmetic progression, $k^{th}$ term is $5k+1$.

**To do:**To find the sum of first $100$ terms.

**Solution:**

Let $a$ be the first term of an AP and $d$ is the common difference.

$\therefore a_k=a+(n-1)d$

Since, $a_k=5k+1$

$a+( k-1)d=5( k-1)+6$

$\Rightarrow a+( k-1)d=6+( k-1)5$

Equating both sides, we get

$a=6$ and $d=5$

$\therefore$ Sum of $100$ terms, $S_{100}=\frac{n}{2}[2a+( n-1)d]$

$=\frac{100}{2}[2\times6+99\times5]$

$=50[12+495]=50( 507)=25, 350$

Thus, the sum of $100$ terms is $25, 350$.

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