If `S_n=n P+(n(n-1))/2Q ,w h e r eS_n` denotes the sum of the first `n` terms of an A.P., then find the common difference.

Options

A. `P+Q`
B. `2P+3Q`
C. `2Q`
D. Q

Answer

D

Explanation

`S_(n)="np"+(n(n-1)Q)/(2)`
We know, `T_(1)=S_(1)" and "T_(2)=S_(2)-S_(1)`
Common difference (d) `=T_(2)-T_(1)`
`:.S_(1)=(1)P+(1(1-1)Q)/(2)=P+0=P`
`S_(2)=(2)P+(2(2-1)Q)/(2)=2P+(2Q)/(2)=2P+Q.`
`:.T_(1)=P,T_(2)=2P+Q-P+Q`
`:."Common difference (d)"=T_(2)-T_(1)`
`=P+Q-P=Q.`