JEE MAIN - Mathematics (2021 - 17th March Morning Shift - No. 6)
The value of $$4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}}$$ is :
2 + $${2 \over 5}\sqrt {30} $$
2 + $${4 \over {\sqrt 5 }}\sqrt {30} $$
5 + $${2 \over 5}\sqrt {30} $$
4 + $${4 \over {\sqrt 5 }}\sqrt {30} $$
Explanation
$$y = 4 + {1 \over {5 + {1 \over y}}}$$
$$ \Rightarrow y = 4 + {y \over {5y + 1}}$$
$$ \Rightarrow 5{y^2} - 20y - 4 = 0$$
$$ \Rightarrow y = {{20 \pm \sqrt {400 + 80} } \over {10}}$$
$$ \Rightarrow y = {{20 \pm 4\sqrt {30} } \over {10}},y > 0$$
$$ \therefore $$ $$y = {{10 + 2\sqrt {30} } \over 5}$$
$$ \Rightarrow y = 4 + {y \over {5y + 1}}$$
$$ \Rightarrow 5{y^2} - 20y - 4 = 0$$
$$ \Rightarrow y = {{20 \pm \sqrt {400 + 80} } \over {10}}$$
$$ \Rightarrow y = {{20 \pm 4\sqrt {30} } \over {10}},y > 0$$
$$ \therefore $$ $$y = {{10 + 2\sqrt {30} } \over 5}$$
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