The infinite root

Calculus Level 3

1 + 2 1 + 3 1 + 4 1 + = ? \sqrt{1 + 2\sqrt{1 + 3 \sqrt{1 + 4\sqrt{1 + \cdots}}}} = \, ?


The answer is 3.

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1 solution

Justin Arun
May 28, 2018

Turn this equation into something like this: F ( x ) = ( 1 + x ) ( 1 + ( x + 1 ) ) F(x)=√(1+x)√(1+(x+1)) Then, square both sides. F ( x ) s q u a r e d = 1 + x F ( x + 1 ) F(x) squared =1+xF(x+1) This will give solution: F ( x ) = x + 1 F(x)=x+1 Setting Function x = 2 x=2 gives 3.

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