The Mathematicians Square[Root]

Geometry Level 1

A Mathematician has a square of sides length $\sqrt { 2 }$

He's very clever and is able to work out the area of the square.

However, he suddenly sees that he can write his answer for the area out in a different way! In the form of:

$\sqrt { x\sqrt { x\sqrt { x\sqrt { x\sqrt { x.... } } } } }$

What is the value of x?

1 solution

Ben Farrar
Aug 10, 2014

The Area=2 (from $\sqrt { 2 } \times \sqrt { 2 } =\sqrt { 4 } =2$ )

Therefore : $2=\sqrt { x\sqrt { x\sqrt { x\sqrt { x\sqrt { x.... } } } } }$

If you then see that the series is embedded within itself you can use the substitution of 2 to make:

$2=\sqrt { x(2) } \\ 2=\sqrt { 2x } \\ 4=2x\\ x=2$