Question

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

Answer 1

Let `x = sqrt(10 + sqrt(10 + sqrt(10 + ...)))`

Then `x = sqrt(10 + x)`

Solving this equation for x:

`x^2 = (sqrt(10 + x))^2`

`x^2 = 10 +x`

`x^2 - x - 10 = 0`

`x = (-b +- sqrt(b^2 - 4ac))/(2a)`

`x = (-(-1) +- sqrt(-1^2 - 4 xx 1 xx -10))/(2 xx 1)`

`x = (1 +- sqrt(41))/2`

However, `x = (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 + ...)))` is `(1 + sqrt(41))/2`.

Hopefully this helps!