8.3 Geometric Sequences and Series Objectives: -Students will recognize, write, and find the nth terms of geometric sequences. -Students will find the nth partial sums of geometric sequences. -Students will find sums of infinite geometric series. -Students will use geometric sequences to model and solve real-life problems.

Definition of Geometric Sequence A sequence is geometric if the ratios of consecutive terms are the same. a 1, a 2, a 3, a 4, …, a n, … is geometric if there is a number r such that r is the common ratio of the geometric sequence

Ex 1) Determine whether the sequence is geometric. If it is, find the common ratio. a) 6, 18, 30, 42, … b) 1, -½, ¼, -⅛, …

The nth Term of a Geometric Sequence The nth term of a geometric sequence has the form a n = a 1 · r n – 1 where r is the common ratio.

Ex 2) Write the first 5 terms of the geometric sequence. Find the common ratio and write the nth term of the sequence as a function of n. a 1 = 64 and a k+1 = ½ a k

Ex 3) Find the nth term of the geometric sequence. Use the table feature of your calculator to verify your answer. a 1 =4, r = ½, n = 10

Ex 4) Find the nth term of the geometric sequence. Use the table feature of your calculator to verify your answer. a 2 = -18, a 5 =2/3, n = 6

The Sum of an Infinite Geometric Sequence If │r│ 1, the series does not have a sum. The Sum of a Finite Geometric Sequence

Ex 5) Find the sum. Use a calculator to verify.

Ex 6) Find the sum. Use a calculator to verify.

Ex 7) Use summation notation to write the sum …

Ex 8) Find the sum of the infinite geometric series, if possible. If not possible, explain why.

Ex 9) Find the sum of the infinite geometric series, if possible. If not possible, explain why.

Ex 10) Find the rational number representation of the repeating decimal. a) b)