Question

If the common ratio of an infinite $G.P$ be less then $\frac{1}{2}$, show that each term will be greater than the sum of all the terms that follow it.

Answer 1

$T_p>T_{p+1}+T_{p+2}+...\infty$

$\therefore {ar}^{p-1}>\frac{a{r}^p}{1-r}$ .....$\left(1\right)$

where $r<1$ or $1-r=+ive$

Multiply the inequality $\left(1\right)$ by $+ive$ quantity $1-r$ and cancel ${ar}^{p-1}$

$\therefore 1-r>r$ or $1>2r$ i.e if $r<\frac{1}{2}$