How can I derive an equation to find the sum of an arithmetic sequence?

This isn't a requirement of many courses but understanding and proving mathematics has order is what makes mathematics unique and enjoyable to many.
Imagine the sum of a sequence with n terms, denoted S_n, which has an initial value, a, and a constant value, d, added on to each term.
For example, my sequence could be:
1, 6, 11, 16... which would make a = 1 and d = 5.
Then S_n = 1 + 6 + 11 + 16 +...
So to be more general,
S_n = a + (a+d) + (a+2d) + ...+ (a+(n-2)d) + (a+(n-1)d)
S_n = (a+(n-1)d) + (a+(n-2)d) +...+ (a+2d) + a [reverse S_n]
Add both the sums together: add the first term to the other first term, then the second to the other second and so on.
2S_n = (2a+(n-1)d) + (2a+(n-1)d) +...+ (2a+(n-1)d) + (2a+(n-1)d)
There are n amounts of (2a+(n-1)d), as there are n terms, so this can be factored out.
2S_n = n(2a + (n-1)d)
=> S_n = 0.5n(2a + (n-1)d)    [divide by 2]