SSC CGL Model Paper 3

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Question : 64 of 100

Marks: +1, -0

The value of

x + √{x^2+√{x^4+√{x^8+√{x^16+...}}}}

$x({√5+2}/2)$
$x({3+√5 }/2)$
$x/{1+√x}$
$x({√5+1}/2)$

Solution:

x + √{x^2+√{x^4+√{x^8+√{x^16+...}}}}

=

x + √{x^2+√{x^4+√{x^8+x^8√{1+...}}}}

=

x + √{x^2+√{x^4+√{x^8(1+ √{ +...)}}}}

=

x + √{x^2+√{x^4+x^4√{ 1+ √1 }}}

=

x + √{x^2+√{x^4(1+√{ 1+....)}}}

=

x + √{x^2+ {x^2√{1+√{ 1+.... }}}

=

x +x √{ 1+√{1+ √{ 1+.... }} }

=

x(1+√{1+√{1+√{1+....)}}}

Now
Let

1 +√{1√{1+√{1+√{1+.... }}}}

= = y (let)
∴

1+√y=y⇒√y=y-1

Squaring both sides

y = y^ 2 + 1 - 2y

⇒

y^2- 3 y + 1 = 0

Then

y={3±√{9-4}}/2= {3±√5 }/2

∵

1 + √{1+√{1+...}}

will be positive and equal to

{3+√5 }/2

∴

x(1 + √{1+√{1+...}})

=x({3+√5 }/2)

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