Generating Formulas for Sums

Given a sequence $a_n$, we denote the sequence of partial sums by $s_n=\sum\limits_{k=1}^n a_k$. There are several approaches that can be used to find sums, and formulas for sums.

Write out each term and add them up. This approach is conceptually easy, but can be tedious.
Use the summation capabilities of a calculator or computer. Many such sums can be handled quickly in this fashion, but beyond the answer, not much is learned about the relationships by doing so.
Algebraically exploiting the pattern in the sequence. We shall give some examples below.
Build a library of partial sums for special cases, to use with properties of sums.

Properties of Sums

As long as the number of terms in the sum is finite, certain properties will hold.

$\quad \sum\limits_{k=1}^n (a_k+b_k) = \sum\limits_{k=1}^n a_k + \sum\limits_{k=1}^n b_k$
$\quad \sum\limits_{k=1}^n c(a_k) = c \sum\limits_{k=1}^n a_k$

The first result is an immediate consequence of the Associative and Commutative Properties of Multiplication. The second result is the Distributive Property. These two results make it easy to manipulate sums of functions of a variable, if a library of special cases is available. Here are a few of those special cases.

$\quad \sum\limits_{k=1}^n c = nc$
$\quad \sum\limits_{k=1}^n k = \dfrac{n(n+1)}{2}$
$\quad \sum\limits_{k=1}^n k^2 = \dfrac{n(n+1)(2n+1)}{6}$
$\quad \sum\limits_{k=1}^n k^3 = \dfrac{n^2(n+1)^2}{4}$
$\quad \sum\limits_{k=1}^n c^k = \dfrac{c^{n+1}-c}{c-1}$

Each of these results can be proven by mathematical induction. The first four of these special cases are particularly useful when polynomials are involved. It should be noted that in each case, the degree of the sum is one more than the exponent of the quantity being summed. The last of these special cases is useful when a formula is exponential, as in the geometric sequence.