How do you find the formula of infinite surds?

Answer 1

x =√k+√k+√k…… ect

x² = k + x

x² - x = k

x² - x + (1/2)² = k + (1/2)² factorise x² - x + (1/2)²

(x + 1/2)² = k + 1/4

(x + 1/2)² = (4k + 1)/4

x + 1/2 = ± √(4k + 1)/2

x = 1/2 ± √(4k + 1)/2

it obviously has to be positive

x = 1/2 + √(4k + 1)/2

x = (1 + √(4k + 1))/2