What is the exact value of sqrt(10+sqrt(10+sqrt(10+...))) ?

Sep 27, 2016

Let $x = \sqrt{10 + \sqrt{10 + \sqrt{10 + \ldots}}}$

Then $x = \sqrt{10 + x}$

Solving this equation for x:

${x}^{2} = {\left(\sqrt{10 + x}\right)}^{2}$

${x}^{2} = 10 + x$

${x}^{2} - x - 10 = 0$

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$x = \frac{- \left(- 1\right) \pm \sqrt{- {1}^{2} - 4 \times 1 \times - 10}}{2 \times 1}$

$x = \frac{1 \pm \sqrt{41}}{2}$

However, $x = \frac{1 - \sqrt{41}}{2}$ is extraneous since it doesn't satisfy the original equation.

Hence, the value of the expression $\sqrt{10 + \sqrt{10 + \sqrt{10 + \ldots}}}$ is $\frac{1 + \sqrt{41}}{2}$.

Hopefully this helps!