Objectives Find terms of a geometric sequence, including geometric means. Find the sums of geometric series.

Vocabulary geometric sequence geometric mean geometric series

Example 1A: Identifying Geometric Sequences Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. 100, 93, 86, 79, ...

Example 1B: Identifying Geometric Sequences Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. 180, 90, 60, 15, ...

Example 1C: Identifying Geometric Sequences Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. 5, 1, 0.2, 0.04, ...

Check It Out! Example 1a Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.

Check It Out! Example 1b Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. 1.7, 1.3, 0.9, 0.5, . . .

Each term in a geometric sequence is the product of the previous term and the common ratio, giving the recursive rule for a geometric sequence. an = an–1r nth term Common ratio First term

Check It Out! Example 2b Find the 9th term of the geometric sequence. 0.001, 0.01, 0.1, 1, 10, . . .

Example 3: Finding the nth Term Given Two Terms Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324. Step 1 Find the common ratio. a5 = a3 r(5 – 3) Use the given terms.

Example 3 Continued Step 2 Find a1. Consider both the positive and negative values for r. an = a1r n - 1 an = a1r n - 1 36 = a1(3)3 - 1 or 36 = a1(–3)3 - 1

Example 3 Continued Step 3 Write the rule and evaluate for a8. Consider both the positive and negative values for r. an = a1r n - 1 an = a1r n - 1 an = 4(3)n - 1 or an = 4(–3)n - 1

values for r when necessary. Caution! When given two terms of a sequence, be sure to consider positive and negative values for r when necessary.

Check It Out! Example 3a Find the 7th term of the geometric sequence with the given terms. a4 = –8 and a5 = –40

Check It Out! Example 3b Find the 7th term of the geometric sequence with the given terms. a2 = 768 and a4 = 48

The indicated sum of the terms of a geometric sequence is called a geometric series. You can derive a formula for the partial sum of a geometric series by subtracting the product of Sn and r from Sn as shown.

Example 5A: Finding the Sum of a Geometric Series Find the indicated sum for the geometric series. S8 for 1 + 2 + 4 + 8 + 16 + ...

Example 5A: Finding the Sum of a Geometric Series Find the indicated sum for the geometric series. S8 for 1 + 2 + 4 + 8 + 16 + ... Step 1 Find the common ratio.

Example 5A Continued Step 2 Find S8 with a1 = 1, r = 2, and n = 8. Sum formula Check Use a graphing calculator. Substitute.

Example 5A Continued Step 2 Find S8 with a1 = 1, r = 2, and n = 8. Sum formula Check Use a graphing calculator. Substitute.

Example 5B: Finding the Sum of a Geometric Series Find the indicated sum for the geometric series.

Example 5B: Finding the Sum of a Geometric Series Find the indicated sum for the geometric series. Step 1 Find the first term.

Example 5B Continued Step 2 Find S6. Check Use a graphing calculator. Sum formula Substitute.

Example 5B Continued Step 2 Find S6. Check Use a graphing calculator. Sum formula Substitute. = 1(1.96875) ≈ 1.97

Check It Out! Example 5a Find the indicated sum for each geometric series. S6 for

Check It Out! Example 5a Find the indicated sum for each geometric series. S6 for Step 1 Find the common ratio.

Check It Out! Example 5a Continued Step 2 Find S6 with a1 = 2, r = , and n = 6.

Check It Out! Example 5a Continued Step 2 Find S6 with a1 = 2, r = , and n = 6. Sum formula Substitute.

Check It Out! Example 5b Find the indicated sum for each geometric series.

Check It Out! Example 5b Find the indicated sum for each geometric series. Step 1 Find the first term.

Check It Out! Example 5b Continued Step 2 Find S6.

Check It Out! Example 5b Continued Step 2 Find S6.

Lesson Quiz: Part I 1. Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. 2. Find the 8th term of the geometric sequence 1, –2, 4, –8, …. 3. Find the 9th term of the geometric sequence with a2 = 0.3 and a6 = 0.00003.

Lesson Quiz: Part II 4. Find the geometric mean of and 18. 5. Find the indicated sum for the geometric series 6. A math tournament begins with 81 students. Students compete in groups of 3, with 1 person from each trio going on to the next round until there is 1 winner. How many matches must be played in order to complete the tournament?