Question

Calculate the sum of $\frac{8}{5},-1,\frac{5}{8},.....$

Answer 1

Number of terms $n=\infty$
Clearly given series is G.P. with common ratio $r=\frac{-5}{8}$ , $\left|r\right|<1$ and first term =$a=\frac{8}{5}$
Formula for sum of $n$ terms of G.P, is given by
$S_n=\frac{a(r^n-1)}{r-1}$
If $n=\infty$ & $\left|r\right|<1$ then formula for total sum is
$S=\frac{a}{1-r}$
Substituting the values, we get
$S=\frac{\frac{8}{5}}{1-\left(\frac{-5}{8}\right)}=\frac{64}{65}$