\(\sum_{i=1}^n a_i = a_1 + a_2 + ... + a_n\)

The symbol is called sigma.

\(\sum_{i=1}^n c = nc\)

\(\sum_{i=1}^n i = \dfrac{n(n+1)}{2}\)

For any numbers \(c, a_1, a_2\) … \(a_n, b_1, b_2\) … \(b_n\), the following equations are true:

\(\sum_{i=1}^n (ca_i) = c\sum_{i=1}^na_i\)

\(\sum_{i=1}^n (a_i + b_i) = \sum_{i=1}^n a_i + \sum_{i=1}^n b_i\)

\(\sum_{i=1}^n (a_i - b_i) = \sum_{i=1}^n a_i - \sum_{i=1}^n b_i\)

\(\sum_{i=1}^n [(i+1)^3 -1^3] = [2^3 - 1^3] + [3^3 - 2^3] + [4^3 - 3^3] + ... + [n^3 - (n-1)^3] + [(n+1)^3-n^3]\)

Notice how the terms cancel each others. The only thing that is left finally is \(-1^3\) from the first term and the \((n+1)^3\) from the last term.

Sums like above where most of the terms cancel out are called telescoping sums.

\(\sum_{i=1}^n i^2 = \dfrac{n(n+1)(2n+1)}{6}\)

\(\sum_{i=1}^n i^3 = \dfrac{n^2(n+1)^2}{4}\)

Dummy or Bound variables in an experession can be changed to a different variable without affecting the value of the expression.