Question

If the sum of 'n' terms of an A.P is $(pn+q{n}^2)$ where p and q are constant find the common difference

Answer 1

$Given$ : $S_n=pn+q{n}^2.......\left(1\right)$
$Substituting$ $n=1,2weget$
$S_1=p\left(1\right)+q(1{)}^2=p+qand$
$S_2=p\left(2\right)+q(2{)}^2=2p+4q$
$Weknowthat$ , $S_1=a$
$S_2=a+d$
$S_2-S_1=(a+d)-\left(a\right)=d.......\left(2\right)$

$S_2-S_1=(2p+4q)-(p+q)....\left(3\right)$

$From\left(1\right)and\left(2\right)$

$p+3q=d$