13 3 arithmetic and geometric series and their sums

13-3 Arithmetic and Geometric Series and Their SumsObjectives:

1. Find the sum of the first n terms of arithmetic or geometric series

What is a Series?The sum of the terms of a sequence.

Finite sequence: 2, 6, 10, 14Finite series: 2 + 6 + 10 + 14

Infinite sequence:Infinite series:

NotationSn represents the sum of n terms of a series.

Explicit definition of Sn

Sn = t1 + t2 + t3 + … + tn

Finite Arithmetic SeriesThe sum of the first n terms of an arithmetic series is:

(The number of terms times the avg. of the first and last terms.)

Example 1Find the sum of the cubes of the first 20 positive integers.

Example 2Find the sum of the first 25 terms of the series 11 + 14 + 17 + 20 + …

You Try!Find the sum of the series:3 + 7 + 11 + … + 99

Finite Geometric SeriesThe sum of the first n terms of a geometric series is:

ExampleFind the sum of the first 10 terms of 2 – 6 + 18 – 54 + ….

You Try!Find the sum of 3 + 9 + 27 + … + 6561