Problem

Write the following sum in sigma notation:

Solution

This sum can be written as

To condense this to sigma notation, we should look for numbers that increase, or follow some sort of pattern, with each successive term. Here, the first multiplier of each term increases, following the sequence 1, 2, 3, ..., n–1. Also, the exponent in each term increases in the same way. So we can use the variable i to represent both the first multiplier and the exponent, and sum as i ranges from 1 to n–1. Using sigma notation, this is

To check that this is correct, we can expand the sum for i = 1, i = 2, i = 3, ..., i = n–1 and hope we get what we started with.

Indeed, we get

More correct answers

There are other correct answers besides this one. You may wish to start the summation at some other value than 1, or to use a different range variable, or to factor terms out of the summation. The following summations are equivalent to the one derived above:

Each of these, when expanded, is equivalent to the original expression: