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# The first term of an A.P. is 5, the last term is 45 and the sum of all its terms is 400. Find the number of terms and the common difference of the A.P.

**Given:**The first term of an A.P. is 5, the last term is 45 and the sum of all its terms is 400.

**To do:**To find the number of terms and the common difference of the A.P.

**Solution:**

First term $a=5$

Last term $l= 45$

Sum of all the terms $S_{n} =400$

let the number of terms of the given A.P. is $n$ and the common differnce is d.

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

$\Rightarrow 400=\frac{n}{2}( 5+45)$

$\Rightarrow 800=50n$

$\Rightarrow n=\frac{800}{50} =16$

And also it is known,

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

on subtituting the values of $a$, $l$ and $n$

$45=5+( 16-1) d$

$\Rightarrow 15d=45-5=40$

$\Rightarrow d=\frac{40}{15} =\frac{8}{3}$

Thus the given A.P. has 16 terms and its common difference is $\frac{8}{3}$.

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