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

Question

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 (A) P + Q (B) P - Q (C) 2 P (D) 2 Q

Tardigrade Admin · Accepted Answer

S1 = P+Q ∴ T1 = P+Q S2 = 2P+4Q ⇒ T1+T2 = 2P+4Q ⇒ T2=2P+4Q -T1 = 2P +4Q-P-Q = P+3Q ∴ common difference = T2-T1 = P+3Q -P-Q = 2Q

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

$P + Q$

$P - Q$

$2 P$

$2 Q$

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

Q. 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

P+Q

P−Q

2P

2Q