For every arithmetic and geometric sequence, there is an associated sum called a series.
For instance, the sequence 1, 4, 7, 10, 13, 16, 19, 22, 25, 28 has the associated series, 1+ 4+ 7+10+13+16+19+22+25+28.

Sigma Notation is a notation used to abbreviate long summations of a given expression.
For instance, the above sum can be written using sigma notation as shown below:

The upper case Greek letter sigma, written as, is used to indicate that a given expression is a sum. The variable n is used to indicate the interval in which the summation will start and end. The 1+3(n-1) is an expression for the nth term of the associatecd sequence.

Example 1
Write each of the following series in expanded form and find the sum.

a)           Solution

b)     Solution

Example 2 Write each of the following using sigma notation. a) 26 + 23 + 20 + 17 + 14 + 11 + 8 + 5     Solution     Write a formula for the nth term of the arithmentic series.          There are 8 terms in the series.     Thus, the series can be written as: .

b) 2 + 10 + 50 + 250 + 1250 + 6250     Solution     Write a formula for the nth term of the geometric series.          There are 6 terms in the series.     Thus, the series can be written as: .