Geometric Sequences & Series Section 8.3 Geometric Sequences & Series

Geometric Sequences & Series A sequence in which a set number is multiplied to each previous term is called a geometric sequence. Definition of Geometric Sequence A sequence is geometric if the ratios of consecutive terms are the same. So, the sequence a1, a2, a3, a4, …, an, … is geometric if there is a number r such that The number r is the common ratio of the sequence.

Geometric Sequences & Series Examples: Determine if each of the following is a geometric sequence. If so, determine the common ratio.

Geometric Sequences & Series How does one find a certain term of a geometric sequence? The nth term of a Geometric Sequence The nth term of a geometric sequence has the form an = a1rn–1 where r is the common ratio of consecutive terms of the sequence.

Geometric Sequences & Series Examples: 1. What are the first five terms of the geometric sequence whose first term is a1 = 2 and whose common ratio is r = 3. 2. Find the 10th term of the geometric sequence whose first term is 8 and whose common ratio is ¼ .

Geometric Sequences & Series Examples: 4. Find the 8th term of the geometric sequence whose third term is 1/25 and whose sixth term is 1/3125. Find a formula for the nth term of the following sequence: 4, 12, 36, 108, 324,…

Geometric Sequences & Series The Sum of a Finite Geometric Sequence The Sum of a Finite Geometric Sequence The sum of the finite geometric sequence a1, a1r, a1r2, a1r3, a1r4, …, a1rn–1 with common ratio r  1 is given by

Geometric Sequences & Series Examples: Find the sum

Geometric Sequences & Series How does one find the sum of an infinite geometric series? The Sum of an Infinite Geometric Series If |r| < 1, then the infinite geometric series a1 + a1r + a1r2 + a1r3 +… + a1rn–1 + … has the sum

Geometric Sequences & Series Example: Find the sum of the infinite geometric series defined by Remember to use the formula for an infinite geometric series. Why?

Geometric Sequences & Series Example: Brandon deposits $100 at the beginning of each month into an account that pays 6% interest compounded monthly. What is the balance at the end of 3 years? Remember the first deposit will earn interest for 36 months, but the last deposit only earns interest for one month which is a1.

Geometric Sequences & Series What you should know: 1. How to find the nth term of a geometric sequence. 2. How to find the partial sum of a geometric sequence (applying the sum formula for a finite geometric series). 3. How to find the sum of an infinite geometric series.