[Coda]

That is... entirely possible. I don't actually know. If the "north" side of the compass needle points to the south pole of a permanent magnet, then you are correct.

What it's saying is if you know the first term of a sequence, and if you know the (constant) distance between subsequent terms, then you can find any term of the sequence using that last line you derived, and you can find the sum of the sequence at any point using the identity you posted.

It should be noted that this follows from theorems you already know:

Σx = n(n+1)/2 -- the sum of the natural numbers
a(b+c) = ab + ac -- the distributive property of multiplication over addition
a + a + ... + a = na -- the additive definition of multiplication

Why?

Well, consider the sum of the integers:
1 + 2 + ... + n = n(n+1)/2

Our life is going to be much easier if we start with zero, though, so let's substitute (n-1) in place of n:
0 + 1 + ... + (n-1) = (n-1)n/2

Multiply it by d:
d(0 + 1 + ... + n) = d(n-1)n/2

Distribute the d:
0 + d + 2d + ... + nd = d(n-1)n/2

Add a₁ to each term, which is the same as adding n copies of a₁:
a₁ + (a₁+d) + (a₁+2d) + ... + (a₁+nd) = d(n-1)n/2 + n(a₁)

But of course, a₁+d is a₂:
a₁ + a₂ + ... + a_n = d(n-1)n/2 + n(a₁)