Question

Calculate the sum: $5+(-5)+5+(-5)+......$
(a) If the number of terms is $10$
(b) If the number of terms is $11$.

Answer 1

Given series $5+(-5)+5+(-5)+......$ is a $GP$ with common ratio = $-1$

Since, we have the sum of first $n$ term is a G.P with common ratio $r$ and first term $a$ is given by
$a\times \frac{(1-r^n)}{1-r}$ when $r<1$ .......(1)

(a) Since $a=5$ and $r=10$
Substitute $a$ and $r$ values in above equation(1)
Therefore, sum = $\frac{\left(5\right)(1-(-1{)}^{10})}{(1-(-1\left)\right)}=\frac{5\times 0}{1+1}=0$

(b) Since $a=5$ and $r=11$
Substitute $a$ and $r$ values in above equation(1),
Therefore, sum = $\frac{5(1-(-1{)}^{11})}{(1-(-1\left)\right)}=\frac{5\times 2}{1+1}=\frac{5\times 2}{2}=5$