Question 1
i) If $x = \left(1 + \frac{1}{n}\right)^n$ and $y = \left(1 + \frac{1}{n}\right)^{n+1}$, show that $y^x = x^y$.
ii) Show that, for all positive integers $n$,
$$
1^2 - 2^2 + 3^2 - 4^2 + \cdots + (-1)^n (n-1)^2 + (-1)^{n+1} n^2 = (-1)^{n+1} (1 + 2 + \cdots + n)
$$
ii) Show that, for all positive integers $n$,
$$
1^2 - 2^2 + 3^2 - 4^2 + \cdots + (-1)^n (n-1)^2 + (-1)^{n+1} n^2 = (-1)^{n+1} (1 + 2 + \cdots + n)
$$