+218 Let \(f(x) = \sqrt{x - \sqrt{x - \sqrt{x - \sqrt{x - \dotsb}}}}.\)Find the largest three-digit value of \(x\) such that \(f(x)\) is an integer.
+128707 Here's my best attempt
Let f(x) = y
And notice that we can write
y = √ [ x - y ] square both sides
y^2 = x - y
y^2 + y = x coplete the square on y
y^2 + y + 1/4 = x + 1/4
(y + 1/2)^2 = [ 4x + 1] / 4 take the positive root
y + 1/2 = √[4x + 1 ] / 2
y = ( √ [ 4x + 1 ] - 1 ) / 2
Let x = (n) (n + 1)
When n (n) (n+1) = x and y
= 1 = 2 = 1
= 2 = 6 = 2
= 3 = 12 = 3
....
= 31 = 992 = 31
The largest three-digit value for x = 992 and this produces y = f(x) = 31
