Question

Let $V_{r}$ done 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)

We have
$v_{r}=\frac{1}{2}\left(2 r^{3}-r^{2}+r\right)$
$v_{r+1}=\frac{1}{2}\left[2(r+1)^{3}-(r+1)^{2}+(r+1)\right] $
$T_{r}=v_{r+1}-v_{r}$
$=(r+1)^{3}-\frac{1}{2}\left[(r+1)^{2}-r^{2}\right]+\frac{1}{2}(1) $
$=2 r^{2}+2 r-1$
$=(r+1)(3 r-1)$
which is a composite number.