Question

If y=1+x+x^2+x^3+.... Infinite so prove x=1-1/y​

Answer 1

Solution:

We have to prove,

$\bullet$ x = 1 - (1/y)

$\bullet$ y = 1 + x + x² + x³ + - - - - ∞

Given series is in geometric progression.

Therefore,

$\implies$ r = common ratio

$\implies$ r = {Tn/(Tn-1)}

$\implies$ r = x/1

$\implies$ r = x

Note: a (first term of series) = 1

The formula for finding infinite sum in geometric series is:

$\implies$ S∞ = a/(1-r)

Hence,

$\implies$ S∞ = 1 / (1-x) = y

$\implies$ (1- x) = 1/y

$\implies$ -x =(1/y) - 1

Taking (-1) on both the sides:

$\bullet$ x = -(1/y)+1

$\bullet$ x = 1-(1/y)

Hence proved.

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Answered by: Niki Swar, Goa❤️