Let `V_r` denote the sum of the first r terms of an arithmetic progression (AP) whose first term is r and the common difference is `(2r-1). Let `T_r=V_(r+1)-V_r-2 and Q_r =T_(r+1)-T_r for r=1,2` `T_r` is always (A) an odd number (B) an even number (C) a prime number (D) a composite num,ber

Options

A. `(1)/(12)n(n+1)(3n^(2)-n+1)`
B. `(1)/(12)n(n+1)(3n^(2)+n+2)`
C. `(1)/(2)n(2n^(2)-n+1)`
D. `(1)/(3)(2n^(3)-2n+3)`

Answer

B

Explanation

`V_(r)=(r )/(2)[(2r+(r-1)(2r-1))]=(1)/(2)(2r^(3)-r^(2)+r)`
`:.sum_(r=1)^(n)V_(r)=(1)/(2)[2sum_(r=1)^(n)r^(3)-sum_(r=1)^(n)r^(3)+sum_(r=1)^(n)r]`
`=(1)/(2)[2((n(n+1))/(2))^(2)-(n(n+1)(2n+1))/(6)+(n(n+1))/(2)]`
`=(1)/(2)n(n+1)(3n^(2)+n+2)`