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. an odd number
B. an even number
C. a prime number
D. a composite number

Answer

D

Explanation

`V_(r+1)-V_(r )=(r+1)^(3)-r^(3)-(1)/(2)[(r+1)^(2)]+(1)/(2)(1)`
`3r^(2)+2r+1`
`:. T_(r )=3r^(2)+2r-1`
`=(r+1)(3r-1)`, which is a composite number.