10 3 Geometric Sequences 10 3 Geometric Sequences

10. 3 Geometric Sequences

10. 3 Geometric Sequences and Series n If we start with a number, a 1, and repeatedly multiply it by some constant, r, then we have a geometric sequence: sequence a 1, a 1 r 2, a 1 r 3, a 1 r 4, …. n n The nth term of a geometric sequence is given by The number r is called the common ratio

10. 3 Geometric Sequences and Series (Example 2) Find the 8 th term of the geometric sequence: 5, 15, 45, … Solution: Use formula, an = a 1 r (n – 1) a 1 = r= n= (Now work Example 3 in text…. )

Geometric Series n n The sum of the terms of a geometric sequence is called a geometric series For example: is a finite geometric series with common ratio r. n What is the sum of the n terms of a finite geometric series?

Deriving a formula for the nth partial sum of a geometric series

10. 3 Geometric Sequences and Series n The sum of the terms of an infinite geometric sequence is an infinite geometric series n n n For some geometric sequences, Sn gets close to a specific number as n gets large. This number becomes the limit of the sum of the infinite geometric sequence. When |r|<1, the limit or sum of an infinite geometric series is given by.

10. 3 Geometric Sequences and Series n You should be able to: n n Identify the common ratio of a geometric sequence, and find a given term and the sum of the first n terms. Find the sum of an infinite geometric series, if it exists.

10. 1 Sequences and Series n You should be able to: n n Find terms of sequences given the nth term. Look for a pattern in a sequence and try to determine a general term. Convert between sigma notation and other notation for a series. Construct the terms of a recursively defined sequence.