Question 81
Prove that for every positive integer n, the expression $A_n$/$B_n$, where
$$\begin{aligned}A_n&=\sqrt{n+\sqrt0}+\sqrt{n+\sqrt1}+\sqrt{n+\sqrt2}+\cdots\\&\quad+\sqrt{n+\sqrt{n^2-1}}+\sqrt{n+\sqrt{n^2}},\\B_n&=\sqrt{n-\sqrt0}+\sqrt{n-\sqrt1}+\sqrt{n-\sqrt2}+\cdots\\&\quad+\sqrt{n-\sqrt{n^2-1}}+\sqrt{n-\sqrt{n^2}},\end{aligned}$$
is independent of n.
$$\begin{aligned}A_n&=\sqrt{n+\sqrt0}+\sqrt{n+\sqrt1}+\sqrt{n+\sqrt2}+\cdots\\&\quad+\sqrt{n+\sqrt{n^2-1}}+\sqrt{n+\sqrt{n^2}},\\B_n&=\sqrt{n-\sqrt0}+\sqrt{n-\sqrt1}+\sqrt{n-\sqrt2}+\cdots\\&\quad+\sqrt{n-\sqrt{n^2-1}}+\sqrt{n-\sqrt{n^2}},\end{aligned}$$
is independent of n.