Nested Radicals

Algebra Level 2

$\large \sqrt{1+\sqrt{1+\sqrt{1+ \cdots}}} = \ ?$
Give your answer approximate to two decimal places.

1.41 1.61

\sqrt{2}

1.51

1 solution

Caleb Townsend
Feb 12, 2015

Substituting $x = \sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}}$ into itself gives $x = \sqrt{1 + x}$ Note the solution is defined as positive since square root is always positive, if not imaginary. Square the equation to get $x^2 = 1 + x$ The only positive solution to this quadratic is the famous $\phi = \frac{1 + \sqrt{5}}{2} \approx \boxed{1.62}$

can you please elaborate your answer more. the process, is it a fixed formula?

Madel Isidro - 5 years, 9 months ago

yes it's a fixed formula know as Shridar Acharya formula in India. it goes like: b+under root(b^2+4ac)/2a where b is the coefficient of 'x' a is the coefficient of 'x^2'and c is the literal coffeint as in '1' in this soln.

Aditya Singh - 5 years, 8 months ago

Nested Radicals

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