If x = 1 + 1 + 1 + ....... , then x = - Vidyadip

MCQ

If $x = \sqrt {1 + \sqrt {1 + \sqrt {1 + .......\infty} } }$, then $x =$

✓
$\frac{{1 + \sqrt 5 }}{2}$
B
$\frac{{1 - \sqrt 5 }}{2}$
C
$\frac{{1 \pm \sqrt 5 }}{2}$
D
None of these

✓

Answer

Correct option: A.

$\frac{{1 + \sqrt 5 }}{2}$

a
(a) $x = \sqrt {1 + \sqrt {1 + \sqrt {1 + .....\infty} } } $

We have $x = \sqrt {1 + x} $

==> ${x^2} = 1 + x\,\,\,\, \Rightarrow {x^2} - x - 1 = 0$

==> $x = \frac{{1 \pm \sqrt {1 + 4} }}{2} = \frac{{1 \pm \sqrt 5 }}{2}$

As $x > 0$, we get $x = \frac{{1 + \sqrt 5 }}{2}$

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