Question

the first term of an infinite GP is 1 and each term is twice the sum of the succeeding is terms then the sum of the series is ​

Answer 1

Answer:

The sum of the infinite  G.P is 3/2

Step-by-step explanation:

The first term of an infinite GP is 1 and each term is twice the sum of the succeeding is terms then the sum of the series is ​

Formula used:

$\text{The sum of the infinite G.P }a, ar,ar^2.............\text{ is}$

$\boxed{S_{\infty}=\frac{a}{1-r}}$

Given:

a=1 and

$t_n=2(t_{n+1}+t_{n+2}+...........)$

$ar^{n-1}=2(ar^n+ar^{n+1}+ar^{n+2}...........)$

$ar^{n-1}=2\:ar^n(1+r+r^2...........)$

$1=2\:r(1+r+r^2...........)$

$1=2\:r(\frac{1}{1-r})$

$1=\frac{2r}{1-r}$

$1-r=2r$

$1=3r$

$\implies\:r=\frac{1}{3}$

The sum of infinite of G.P is

$=\frac{a}{1-r}$

$=\frac{1}{1-\frac{1}{3}}$

$=\frac{1}{\frac{3-1}{3}}$

$=\frac{1}{\frac{2}{3}}$

$=\frac{3}{2}$