Question

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

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

Answer 1

The correct option is D $-0.5$

To calculate the average of the given series upto 300 terms , the given sequence can be written as
(1, 3, 5, 7,....... 299) + (-2, -4, -6, -8, .........-300)
The given expression is in AP form
The sum of the series is given by
Sn = $\frac{n}{2}\times \left(\right(2a+(n-1)d)$
For the given series,
$Sn=S{n}_1+S{n}_2$
$S{n}_1$ = $\frac{150}{2}\times \left(\right(2+(150-1)2)$
= $75\times (2+298)$
= $75\times 300$
$S{n}_2$ = $\frac{150}{2}\times \left(\right(-4+(150-1)(-2))$
= $75\times (-4-298)$
= $75\times (-302)$
$\therefore$ Sn = $75\times 300$ +$75\times (-302)$
Sn = $75(300-302)=75\times (-2)$
Sn = -150
Now, the average of the series is $\frac{Sn}{300}=\frac{-150}{300}$
= -0.5