Geometric Sequences

A Geometric Sequence is a sequence of numbers in which each term is formed by multiplying the previous term by the same number or expression. The consecutive terms have a common ratio. 1, 3, 9, 27, 81, 243, ... The terms have a common ratio of 3.

Geometric Sequence Example Is the following sequence geometric? 4, 6, 9, 13.5, 20.25, 30.375… Yes, the common ratio is 1.5

Geometric Sequence To find any term in a geometric sequence, use the formula an = a1 rn–1 where r is the common ratio.

Example Find the twelfth term of the geometric sequence whose first term is 9 and whose common ratio is 1.2. an = a1 rn–1 a1 = 9 r = 1.2 a9 = 9 • 1.211 a12 = 66.87 To find the sum of a geometric series, we can use summation notation.

Example Which can be simplified to:

Evaluate the sum of: Convert this to = 7.49952

Series

Series

Series Definition: A series is a partial sum of the first n terms of a sequence. General term: nth partial sum: Sn = . nth partial sum of arithmetic sequence: Example: nth partial sum of an = -1 + 5n. Sn =

Series nth partial sum of geometric sequence: Sum of an infinite geometric sequence: If |r|<1, If |r| 1, a geometric series has no infinite sum. Example:

Series Product notation: