Question

First term and the sum of an infinite G.P are 2 and 3 respectively. Find the common ratio and sum upto n terms

Answer 1

SOLUTION

GIVEN

First term and the sum of an infinite G.P are 2 and 3 respectively.

TO DETERMINE

The common ratio and sum upto n terms

EVALUATION

Let a be the First term and r be the Common ratio

Then sum of infinite number of terms

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

So by the given condition

$\displaystyle \sf{ \frac{a}{1 - r} = 3 }$

$\displaystyle \sf{ \implies \: \frac{2}{1 - r} = 3 }$

$\displaystyle \sf{ \implies \: 3 - 3r = 2}$

$\displaystyle \sf{ \implies \: 3r = 1}$

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

Hence the required common ratio

$\displaystyle \sf{ = \frac{1}{ 3} }$

Now sum of the first n terms

$\displaystyle \sf{ = a \times \frac{1 - {r}^{n} }{1 - r} }$

$\displaystyle \sf{ = 2\times \frac{1 - { \big( \frac{1}{3} \big)}^{n} }{1 - \frac{1}{3} } }$

$\displaystyle \sf{ = 2\times \frac{1 - { \big( \frac{1}{3} \big)}^{n} }{ \frac{2}{3} } }$

$\displaystyle \sf{ = 3\times \Bigg[ 1 - { \bigg( \frac{1}{3} \bigg)}^{n}\Bigg] }$

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