Question

The first term of an A.P. of consecutive integers is $p$${}^2$ $+1.$The sum $2p+1$ terms of this series can be expressed as

• A. $(p+1)$${}^2$
• B. $(2p+1)(p^2+p+1)$
• C. $(p+1)$${}^3$
• D. $p$${}^3$$+(p+1)$${}^3$

Answer 1

Answer: (B)

$(2p+1)(p^2+p+1)$

The first term of AP is set of consecutive integers, then common difference is $d=1$ & $a=p^2+1$

Here we use the formula for sum of n terms of an AP as given below:

$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

$\therefore$ The sum of $(2p+1)$, $S_{2p+1}$ =$\frac{2p+1}{2}\left[2\right(p^2+1)+(2p+1-1)\times 1]$

$=\frac{2p+1}{2}[2p^2+2+2p]$

$=(2p+1)(p^2+p+1)$