Question

Write the next two terms of the given sequence $p,0,-p,$ ............ to $p$ terms.

Answer 1

We know the sum of $n$ terms of an A.P with first term $a$ and the common difference $d$ is:

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

Here, it is given that the first term is $a=p$, common difference is $d=0-p=-p$ and $n=p$, therefore, the sum is:

$S_n=\frac{p}{2}\left[\right(2\times p)+(p-1\left)\right(-p\left)\right]=\frac{p}{2}(2p-p^2+p)=\frac{p}{2}(3p-p^2)=\frac{p^2}{2}(3-p)$

Hence, the sum is $\frac{p^2}{2}(3-p)$.