Nested floors

Algebra Level 3

Solve: $\sqrt{\lfloor \sqrt{8} \rfloor+\sqrt{\lfloor \sqrt{8} \rfloor+\sqrt{\lfloor \sqrt{8} \rfloor+\sqrt{\lfloor \sqrt{8} \rfloor+\sqrt{\lfloor \sqrt{8} \rfloor+{. . .}}}}}}$

The answer is 2.

1 solution

A Former Brilliant Member
May 12, 2016

$\lfloor \sqrt8 \rfloor=2$ .

The expression is equivalent to $\sqrt{2 +\sqrt{2+\sqrt{2 \ldots}}}$

Let $x=\sqrt{2 +\sqrt{2+\sqrt{2 \ldots}}}$ .

We have $\sqrt{2+x}=x$ or $x^2-x-2$ .

Solving the quadratic equation and taking only the positive root of $x$ (as the radical symbol of a number is always positive) we get $x=\boxed{2}$ .

Nice solution. To be rigorous you should also prove the convergence of the expression.

展豪張 - 5 years, 1 month ago

Nested floors

The answer is 2.

1 solution

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