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 Sn, 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 Sn = 1 + 6 + 11 + 16 +...
So to be more general,
Sn = a + (a+d) + (a+2d) + ...+ (a+(n-2)d) + (a+(n-1)d)
Sn = (a+(n-1)d) + (a+(n-2)d) +...+ (a+2d) + a [reverse Sn]
Add both the sums together: add the first term to the other first term, then the second to the other second and so on.
2Sn = (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.
2Sn = n(2a + (n-1)d)
=> Sn = 0.5n(2a + (n-1)d) [divide by 2]
