Q

If the sum of n terms of an A.P. is (p n plus q n raised to 2), where p and q are constants, find the common difference

8.  If the sum of n terms of an A.P. is $( pn + qn ^ 2 )$ , where p and q are constants,  find the common difference

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If the sum of n terms of an A.P. is $( pn + qn ^ 2 )$,

$S_n =\frac{n}{2}[2a+(n-1)d]$

$\Rightarrow \, \, \frac{n}{2}[2a+(n-1)d]=pn+qn^2$

$\Rightarrow \, \, \frac{n}{2}[2a+nd-d]=pn+qn^2$

$\Rightarrow \, \, an+\frac{n^2}{2}d-\frac{nd}{2}=pn+qn^2$

Comparing coefficients of $n^2$ on both side , we get

$\frac{d}{2}=q$

$\Rightarrow \, \, d=2q$

The common difference of AP is 2q.

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