Question

What is the average of the first $300$ terms of the given sequence?
$1,-2,3,-4,5,-6,.....,n.(-1{)}^{n+1}$

• A. $-1$
• B. $0.5$
• C. $0$
• D. $-0.5$

Answer 1

The correct option is D $-0.5$

Avg $=\frac{[1+3+5+7+...+(2n-1\left)\right]-[2+4+6+...+2n]}{n}$

Here $n=300$

No. of even terms $=150$

No. of odd terms $=150$
Now,

$1+3+5+7+...+150terms{S}_n=\frac{n}{2}[2a+(n-1\left)d\right]S_{n1}=\frac{150}{2}[2\times 1+149\times 2]=22500$

and $2+4+6+8+....+150terms{S}_n=\frac{n}{2}[2a+(n-1\left)d\right]S_{n2}=\frac{150}{2}[2\times 2+149\times 2]=22650$

Average $=\frac{S_{n1}-S_{n2}}{300}=\frac{22500-22650}{300}=-0.5$