If the sum of n terms of an A.P. is (pn + qn^ | vedclass

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(D) The sum of $n$ terms of an $A.P.$ is given by $S_n = pn + qn^2$.The first term $a_1$ is $S_1 = p(1) + q(1)^2 = p + q$.The sum of the first two ter

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$2q$

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(D) The sum of $n$ terms of an $A.P.$ is given by $S_n = pn + qn^2$.The first term $a_1$ is $S_1 = p(1) + q(1)^2 = p + q$.The sum of the first two terms $S_2$ is $p(2) + q(2)^2 = 2p + 4q$.The second term $a_2$ is $S_2 - S_1 = (2p + 4q) - (p + q) = p + 3q$.The common difference $d$ is $a_2 - a_1 = (p + 3q) - (p + q) = 2q$.Alternatively,comparing $S_n = \frac{d}{2}n^2 + (a - \frac{d}{2})n$ with $S_n = qn^2 + pn$,we get $\frac{d}{2} = q$,which implies $d = 2q$.

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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