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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
The value of $3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}$
Solution
Let $x = 4 + \frac{1}{3 + \frac{1}{4 + \cdots}}$Now,
$x = 4 + \frac{1}{3 + \frac{1}{x}}$
$= 4 + \frac{1}{\frac{3x+1}{x}} = 4 + \frac{x}{3x+1}$
$\Rightarrow x(3x+1) = 4(3x+1) + x$
$\Rightarrow 3x^2 + x = 12x + 4 + x$
$\Rightarrow 3x^2 - 12x - 4 = 0$
$\Rightarrow x = \frac{12 + \sqrt{144 + 48}}{6} = \frac{12 + 8\sqrt{3}}{6}$
$\Rightarrow x = 2 + \frac{4\sqrt{3}}{3}$
Now,
$3 + \frac{1}{x} = 3 + \frac{1}{2 + \frac{4\sqrt{3}}{3}}$
$= \frac{3}{2} + \sqrt{3}$
Final Answer:
$(a) 1.5 + \sqrt{3}$
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