Direct Proof of Arithmetic Series Summation
Provide a detailed proof that the sum of the first \(n\) terms of an arithmetic sequence with first term \(a\) and common difference \(d\) is given by \(S_n = \frac{n}{2}[2a + (n-1)d]\).
Arithmetic Sequence Definition and Common Difference
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant difference to the previous term. Given that the first term of an arithmetic sequence is 3 and the fourth term is 15, find the common difference.