Geometric Sequence and Arithmetic Sequence

I learned Geometric Sequence and Arithmetic Sequence in my senior high school. And as a programmer I found them very useful when calculating the time complex of an algorithm. But the theorem related to them are easily forgotten, so I tried to prove them in order to memorize them better.

Sum of Geometric Sequence

A sequence of items:
$a_1, a_2, a_3, ..., a_n$
where
$a_k = a_{k-1}*r$
(r is called common ratio) is called geometric sequence. To get sum of which
$S_n = a_1 + a_2 + a_3+ ...+ a_n$

We first have:

$\begin{aligned} (1)S_n &= a_1 + a_1*r^1 + a_1 * r^2+ ...+ a_1 * r^{n-1} \\ (2) S_{n+1} &= a_1 + a_1*r^1 + a_1*r^2+ ...+ a_1 * r^{n-1} + a_1 * r^n = S_n + a_1 * r^n \\ (3) S_n * r &= a_1*r^1 + a_1*r^2+ ...+ a_1 * r^{n-1} + a_1 * r^n \end{aligned}$

(2) - (1) will give us:
$(4) S_{n+1} - S_n = a_1 * r^n$

(2) - (3) will give us:
$(5) S_{n+1}~ - S_n * r = a_1$

That is two variables in two equations, (4) - (5) will give us
$\begin{aligned} S_n * r - S_n &= a_1 * r^n - a_1 \\ S_n &= (a_1 * r^n - a_1) / (r - 1) \\ &= a_1(r^n - 1) / (r-1) \end{aligned}$

Sum of Arithmetic Sequence

A sequence of items:
$a_1, a_2, a_3, ..., a_n$
where
$a_k = a_{k-1} + d$
(d is called common difference) is called arithmetic sequence. To get sum of which
$S_n = a_1 + a_2 + a_3+ ...+ a_n$

We have:
$(1) S_n = a_1 + a_2 + a_3 + ... + a_n \\ (2) S_n = a_n + a_{n-1} + a_{n-2} + ... + a_1$

(1) + (2) will give us:
$(3) 2 * S_n = (a_1 + a_n) + (a_2 + a_{n-1}) + (a_3 + a_{n-2}) + ... + (a_n + a_1)$
As
$\begin{aligned} a_k &= a_{k-1} + d \\ &= a_{k-2} + d * 2 \\ &= ... \\ &= a_1 + d * (k-1) \end{aligned}$

we have:
$a_k + a_l = 2 * a_1 + d(k+l-2)$

That is as lone as$k+l$is the same,$a_k + a_l$will be the same. So (3) can be expand as
$\begin{aligned} 2 * S_n &= (a_1 + a_n) * n \\ S_n &= (a_1 + a_n) * n / 2 \end{aligned}$