Question

Let $V_r$ denote the sum of the first ' $r$ ' terms of an arithmetic progression (A.P.) whose first term is ' $r$ ' and the common difference is $(2 r -1)$.
Let $T _{ r }= V _{ r +1}- V _{ r }-2$ and $Q _{ r }= T _{ r +1}- T _{ r }$ for $r =1,2, \ldots$
$ T_r$ is always

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

Answer 1

Answer: (D)

$V _{ r } =\frac{ r }{2}\left(2 r ^2- r +1\right)= r ^3-\frac{1}{2} r ^2+\frac{ r }{2} $
$T _{ r } =\left(( r +1)^3- r ^3\right)-\frac{1}{2}\left(( r +1)^2- r ^2\right)+\frac{1}{2}( r +1- r )-2$
$=\left( r ^2+2 r +1+ r ^2+ r ^2+ r \right)-\frac{1}{2}(2 r +1)+\frac{1}{2}-2$
$T _{ r } =3 r ^2+3 r +1- r -2=3 r ^2+2 r -1$
$T _{ r } =3 r ^2+3 r - r -1=(3 r -1)( r +1) \Rightarrow \text { (D) }$