Essential Questions Geometric Sequences and Series How do we find the terms of an geometric sequence, including geometric means? How do we find the sums of geometric series? Holt McDougal Algebra 2 Holt Algebra 2

Serena Williams was the winner out of 128 players who began the 2003 Wimbledon Ladies’ Singles Championship. After each match, the winner continues to the next round and the loser is eliminated from the tournament. This means that after each round only half of the players remain.

The number of players remaining after each round can be modeled by a geometric sequence. In a geometric sequence, the ratio of successive terms is a constant called the common ratio r (r ≠ 1) . For the players remaining, r is .

Recall that exponential functions have a common ratio. When you graph the ordered pairs (n, an) of a geometric sequence, the points lie on an exponential curve as shown. Thus, you can think of a geometric sequence as an exponential function with sequential natural numbers as the domain.

Identifying Geometric Sequences Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. 1. 100, 93, 86, 79, . . . Diff. It could be arithmetic, with d = –7. Ratio 2. 180, 90, 60, 15, . . . Diff. It is neither. Ratio

Identifying Geometric Sequences Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. 3. 5, 1, 0.2, 0.04, . . . Diff. It could be geometric, with r = 0.2. Ratio 4. 1.7, 1.3, 0.9, 0.5, . . . Diff. It could be arithmetic, with d = –0.4. Ratio

The nth term of an geometric sequence is where a1 is the first term and r is the common ratio. General Rule for Geometric Sequence To state a recursive rule of an geometric sequence, state the first term and the rule in the following form: where a1 is the first term and r is the common ratio. Recursive Rule for Geometric Sequence

Finding the nth Term Given a Geometric Sequence Write a rule for the nth term of the geometric sequence. Then find a7.

Finding the nth Term Given a Geometric Sequence Write a rule for the nth term of the geometric sequence. Then find a7.

Finding the nth Term Given a Geometric Sequence Write a rule for the nth term of the geometric sequence. Then find a7.

Finding the nth Term Given Two terms of a Sequence Write a rule for the nth term of the geometric sequence. Then find a8.

Finding the nth Term Given Two terms of a Sequence Write a rule for the nth term of the geometric sequence. Then find a8.

Writing a Recursive Rule for an Geometric Sequence Write the recursive rule for the nth term of the sequence.

Writing a Recursive Rule for an Geometric Sequence Write the recursive rule for the nth term of the sequence.

Lesson 5.3 Practice A