QJEEM21Mada22: JEE Main 2021 Mathematics - Class 11 JEEM21Mada22 Solution

Question

Verified Solution • 2025–26 Session moderate

The value of

3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}

is equal to

Accepted Answer

Let

x = 3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}

So,

x = 3 + {1 \over {4 + {1 \over x}}} = 3 + {1 \over {{{4x + 1} \over x}}}

\Rightarrow (x - 3) = {x \over {(4x + 1)}}

\Rightarrow (4x + 1)(x - 3) = x

\Rightarrow 4{x^2} - 12x + x - 3 = x

\Rightarrow 4{x^2} - 12x - 3 = 0

x = {{12 \pm \sqrt {{{(12)}^2} + 12 \times 4} } \over {2 \times 4}} = {{12 \pm \sqrt {12(16)} } \over 8}

= {{12 \pm 4 \times 2\sqrt 3 } \over 8} = {{3 \pm 2\sqrt 3 } \over 2}

x = {3 \over 2} \pm \sqrt 3 = 1.5 \pm \sqrt 3

.

But only positive value is accepted

So, x =

1.5 + \sqrt 3

Step-by-Step Verified Solution