# Geometric Sequences A Geometric Sequence is n a

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Geometric Sequences

A Geometric Sequence is n 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… n Yes, the common ratio is 1. 5

Geometric Sequence n To find any term in a geometric sequence, use the formula an = a 1 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 = a 1 rn– 1 a 1 = 9 r = 1. 2 a 9 = 9 • 1. 211 a 12 = 66. 87 n To find the sum of a geometric series, we can use summation notation.

Example Which can be simplified to:

Evaluate the sum of: n Convert this to = 7. 49952

Series

Series

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

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

Series n Product notation: