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  4. If the sum of first n terms of an A.P. is Pn + Qn2 where P and Q are constants, then common difference of A.P. will be

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Q. If the sum of first n terms of an $A.P.$ is $Pn + Qn^{2}$ where $P$ and $Q$ are constants, then common difference of $A.P.$ will be

Sequences and Series

Solution:

$S_{1} = P+Q$
$ \therefore T_{1} = P+Q $
$ S_{2} = 2P+4Q $
$ \Rightarrow T_{1}+T_{2} = 2P+4Q$
$\Rightarrow T_{2}=2P+4Q -T_{1} = 2P +4Q-P-Q$
$ = P+3Q $
$ \therefore $ common difference $= T_{2}-T_{1}$
$ = P+3Q -P-Q = 2Q $