Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 1 / d

The iterated function is $f^{(n)}(x)=\sqrt{x^2+n}$ for every positive integer $n$. The exponent here denotes repeated function composition, not an ordinary power or a derivative.

For the base case $n=1$, the formula is the definition. If it holds at $n$, thenThis proves the claim by mathematical induction, for every real input $x$.