Mean and Variance of the Discrete Uniform Distribution

Let $X$ be uniformly distributed on $\{1, 2, \ldots, n\}$, so $P(X = k) = 1/n$ for each $k$.

(a) Derive $E[X]$ in closed form.

(b) Derive $E[X^2]$ using the identity $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$.

(c) Obtain $\text{Var}(X)$.

(d) Evaluate numerically for a fair six-sided die ($n = 6$).