Question

what is the value of X

Answer 1

here is your answer

Answer 2

If √[ x + √x + √x +........= √[ x √x √x.......

then ,what is the value of x ?

solution:
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since, both sides of given expression are of infinite series so then

according to the properties of infinite series,

property 1 :
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√[p + √p + √p + ...... = [√(4p + 1 ) + 1 ] / 2

property 2:
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√[ p √p √p ............... = p

here , we have p = x

so then ,

√[x + √x + √x +.. = [√( 4x + 1 ) + 1 ]/ 2 --(1)

√[ x √x √x ...... = x ------(2)

according to the question,

√[ x + √x + √x +........ = √[ x √x √x ...........

[√(4x + 1 ) + 1 ] / 2 = x

√( 4x + 1 ) + 1 = 2x

√(4x + 1 ) = ( 2x - 1 )

4x + 1 = ( 2x - 1 )^2

use identity:
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(a - b)^2 = a^2 + b^2 - 2ab (in R.H.S) and

solve for 'x':
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4x + 1 = 4x^2 + 1 - 4x

4x + 1 - 1 + 4x = 4x^2

4x^2 = 8 x

4x = 8

x = 2

Answer: value of x = 2

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