# The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.

**Given: **

The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442.

**To do: **

We have to find the common difference of the A.P.

**Solution:**

Let the number of terms of the given A.P. be $n$, first term be $a$ and the common differnce be $d$.

First term $a=2$

Last term $l= 50$

Sum of all the terms $S_{n} =442$

We know that,

Sum of the $n$ terms$ S_{n} =\frac{n}{2}( a+l)$

$\Rightarrow 442=\frac{n}{2}( 2+50)$

$\Rightarrow 442=n(26)$

$\Rightarrow n=\frac{442}{26} =17$

Also,

$l=a+( n-1) d$

Therefore,

On subtituting the values of $a$, $l$ and $n$, we get,

$50=2+( 17-1) d$

$\Rightarrow 16d=50-2=48$

$\Rightarrow d=\frac{48}{16} = 3$

**Hence, the common difference of the given A.P. is $3$. **

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