Summation Notation

A simple method for indicating the sum of a finite (ending) number of terms in a sequence is the summation notation. This involves the Greek letter sigma, Σ. When using the sigma notation, the variable defined below the Σ is called the index of summation. The lower number is the lower limit of the index (the term where the summation starts), and the upper number is the upper limit of the summation (the term where the summation ends). Consider

equation

This is read as “the summation of (2 k + 3) as k goes from 2 to 7.” The replacements for the index are always consecutive integers.

equation

Example 1

Write out the terms of the following sums; then compute the sum.

  1. equation

  2. equation

  3. equation

  1. equation

  2. equation

  3. equation

Example 2

Use sigma notation to express each series.

  1. 8 + 11 + 14 + 17 + 20

  2. equation

  1. 8 + 11 + 14 + 17 + 20

    This is an arithmetic series with five terms whose first term is 8 and whose common difference is 3. Therefore, a 1 = 8 and d = 3. The nth term of the corresponding sequence is

    equation

    Since there are five terms, the given series can be written as

    equation

  2. equation

    This is a geometric series with six terms whose first term is equation and whose common ratio is equation. Therefore, equation and equation. The nth term of the corresponding sequence is

    equation

    Since there are six terms in the given series, the sum can be written as equation