Question

in an infinite GP first term is 1 and sum is 2 the sum of square of term is​

Answer 1

Answer:- 4/3

Step-by-step explanation:

Given:-

First term of an infinite G.P (a) = 1

Sum of the terms in a infinite G.P(S_∞) = 2

To Find :-

Sum of the squares of terms in a infinite G.P.

How To Do :-

As they given the value of 'a'(first term) and 'S_∞'(Sum of the terms in a infinite G.P) by using the formula of Sum of the terms in a infinite G.P , we need to find the value of common ratio(r). After obtaining the value of 'r' we need to substitute in the formula of Sum of the squares of terms in aa infinite G.P.

Formula Required :-

Sum of the terms in a infinite G.P(S_∞) :-$=\dfrac{a}{1-r}$

Sum of the squares of terms in a infinite G.P :-$=\dfrac{a^2}{1-r^2}$

Solution :-

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

[ ∴ S_∞ = a/(1 - r) ]

2(1 - r) = 1

2 - 2r = 1

2 - 1 = 2r

2r = 1

r = 1/2

∴ Common ratio = r = 1/2

Sum of the squares of terms in a infinite G.P :-

$=\dfrac{(1)^2}{1-\left(\dfrac{1}{2}\right)^2}$

$=\dfrac{1}{1-\dfrac{1}{4}}$

$=\dfrac{1}{\dfrac{4-1}{4}}$

$=\dfrac{1}{\dfrac{3}{4}}$

$=\dfrac{4}{3}$

∴ Sum of the squares of terms in a infinite G.P = 4/3.