Question

Write the next two terms of the given sequence $3a,a,-a,$ .......... to $a$ 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 $a=3a$, common difference is $d=a-3a=-2a$ and $n=a$, therefore, the sum is:

$S_n=\frac{a}{2}\left[\right(2\times 3a)+(a-1\left)\right(-2a\left)\right]=\frac{a}{2}(6a-2a^2+2a)=\frac{a}{2}(8a-2a^2)=\frac{2a(4a-a^2)}{2}=a(4a-a^2)=4a^2-a^3$

Hence, the sum is $4a^2-a^3$.