Question

sum of GP upto infinity ​

Answer 1

Given :

Sequence,

$\bigg( \:1 \: + \: \dfrac{1}{ 3} \bigg) \: + \: \bigg( \: \dfrac{1}{2} \: + \: \dfrac{1}{ {3}^{2} } \bigg) \: + \: \: \bigg( \: \dfrac{1}{ {2}^{2} } \: + \: \dfrac{1}{ {3}^{3} } \bigg). \: . \: . \: . \: \infty$

To find :

Sum upto infinity

Formula used :

Sum of infinite term of G.P. is given by ,

$\sf \: S_{\infty}\:=\: \dfrac{a}{1-r}$

Here,

• a = 1st term
• r = common ratio

Solution :

The given sequence is combination of two G.P.

So we can rewrite it as -

$\bigg( \: \dfrac{1}{3} \: + \: \dfrac{1}{ {3}^{2} } \: + \:\dfrac{1}{ {3}^{3} } . \: . \: . \: \infty \bigg) \: + \: \bigg( \: 1 \: + \: \dfrac{1}{ 2 } \: + \:\dfrac{1}{ {2}^{2} } . \: . \: . \: \infty \bigg)$

So we can find sum of both G.P. individually and then add them

• For first G.P.

a₁ = 1/3

r₁ = 1/3

For 2nd G.P.

a₂ = 1

r₂ = 1/2

Therefore

$\sf \implies \: S_{\infty}\:=\:S_{\infty \: 1}\: + \: S_{\infty \: 2}\: \\ \\ \sf \implies \: S_{\infty}\:=\: \dfrac{a_1}{1 - r_1 } \: + \: \dfrac{a_2}{1 - r_2} \\ \\ \\ \sf \implies \: S_{\infty}\:=\: \dfrac{ \dfrac{1}{3} }{1 - \dfrac{1}{3} } \: + \: \dfrac{1}{1 - \dfrac{1}{2} } \\ \\ \\ \sf \implies \: S_{\infty}\:=\: \dfrac{ \dfrac{1}{3} }{ \dfrac{(1 \times 3) - (1 \times 1)}{3} } \: + \: \dfrac{1}{\dfrac{(1 \times 2) - (1 \times 1)}{2} } \\ \\ \\ \sf \implies \: S_{\infty}\:=\: \dfrac{ \dfrac{1}{3} }{ \dfrac{(3 - 1)}{3} } \: + \: \dfrac{1}{\dfrac{(2 - 1)}{2} } \\ \\ \\ \sf \implies \: S_{\infty}\:=\: \dfrac{ \: \: \dfrac{1}{3} \: \: }{ \dfrac{2}{3} } \: + \: \dfrac{ \: \: 1 \: \: }{\dfrac{1}{2} } \\ \\ \\ \sf \implies \: S_{\infty}\:=\: \: \: \bigg( \: \dfrac{1}{3} \: \: \div \dfrac{2}{3} \bigg) \: + \: \bigg( \: 1 \: \div \: \dfrac{1}{2} \bigg) \\ \\ \\ \sf \implies \: S_{\infty}\:=\: \: \: \bigg( \: \dfrac{1}{3} \: \: \times \dfrac{3}{2} \bigg) \: + \: \bigg( \: 1 \: \times \: \dfrac{2}{1} \bigg) \\ \\ \sf \implies \: S_{\infty}\:=\: \: \: \bigg( \: \dfrac{ 3}{6} \: \: \bigg) \: + \: \bigg( \: 2 \bigg) \\ \\ \sf \implies \: S_{\infty}\:=\: \: \: \dfrac{ 1}{2} \: + \: 2$

$\sf \implies \: S_{\infty}\:=\: \: \: \dfrac{ (1 \times 1) + (2 \times 1)}{2}$

$\sf \implies \: S_{\infty}\:=\: \: \: \dfrac{ 1 + 4}{2}$

$\sf \implies \: S_{\infty}\:=\: \: \: \bold{\dfrac{ 5}{2} }$

ANSWER : 5/2