Interpret discrete expected value problem?

Question

An $n$-faced fair die, marked with $1,2,3,...,n$ is rolled. What are the expected number of dots of the landed face?

I understand by my previous question that this will be proportional probability. How do I calculate $E[X]$ given these circumstances.

Given formula:

$$E[X]=\sum_{x, p(x)>0}xp(x)$$

Answer 1

Let $X$ be the value of the dice roll. We are given that $X$ is uniformly distributed on $\{1, 2, \dotsc,n\}$ i.e. $P(X=k)=1/n$ for $k=1,\dotsc, n$. Observe that $n+1-X$ has the same distribution as $X$. In particular $$ E(n+1-X)=E(X) $$ Solve for $E(X)$ using the fact that expectation is linear.

Answer 2

Hopefully the following can help you answer the question:

Since it is fair, we know that $$p(x)=\begin{cases} \frac1n & &, x \in \{ 1, \ldots, n\} \\ 0 & &, \text{Otherwise}\end{cases}$$

Also, the keyword " arithmetic series" might be helpful for this particular question.