Geometric Sequences and Series 9-4 Geometric Sequences and Series Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

Warm Up Simplify. 1. 2. 3. (–2)8 4. Solve for x. 5. 96 Evaluate. 256

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

Vocabulary geometric sequence geometric mean geometric series

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.

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, ... 100, 93, 86, 79 Differences –7 –7 –7 Ratios 93 86 79 100 93 86 It could be arithmetic, with d = –7.

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, ... 180, 90, 60, 15 Differences –90 –30 –45 3 Ratios 1 1 1 2 4 It is neither.

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, ... 5, 1, 0.2, 0.04 Differences –4 –0.8 –0.16 5 Ratios 1 1 1 It could be geometric, with

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

You can also use an explicit rule to find the nth term of a geometric sequence. Each term is the product of the first term and a power of the common ratio as shown in the table. This pattern can be generalized into a rule for all geometric sequences.

Example 2: Finding the nth Term Given a Geometric Sequence Find the 7th term of the geometric sequence 3, 12, 48, 192, .... Step 1 Find the common ratio. r = a2 a1 12 3 = 4 =

Example 2 Continued Step 2 Write a rule, and evaluate for n = 7. an = a1 r n–1 General rule Substitute 3 for a1,7 for n, and 4 for r. a7 = 3(4)7–1 = 3(4096) = 12,288 The 7th term is 12,288.

Check Extend the sequence. a4 = 192 Given a5 = 192(4) = 768 a6 = 768(4) = 3072 a7 = 3072(4) = 12,288 

Check It Out! Example 2a Find the 9th term of the geometric sequence. Step 1 Find the common ratio.

Check It Out! Example 2a Continued Step 2 Write a rule, and evaluate for n = 9. an = a1 r n–1 General rule Substitute for a1, 9 for n, and for r.  

Check It Out! Example 2a Continued Check Extend the sequence. Given a6 = a7 = a8 = a9 =

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. a5 = a3 r2 Simplify. Substitute 324 for a5 and 36 for a3. 324 = 36r2 9 = r2 Divide both sides by 36. 3 = r Take the square root of both sides.

Example 3 Continued Step 2 Find a1. Consider both the positive and negative values for r. an = a1r n - 1 an = a1r n - 1 General rule 36 = a1(3)3 - 1 or 36 = a1(–3)3 - 1 Use a3 = 36 and r = 3. 4 = a1 4 = a1

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 General rule an = 4(3)n - 1 or an = 4(–3)n - 1 Substitute a1 and r. a8 = 4(3)8 - 1 a8 = 4(–3)8 - 1 Evaluate for n = 8. a8 = 8748 a8 = –8748 The 8th term is 8748 or –8747.

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 Step 1 Find the common ratio. a5 = a4 r(5 – 4) Use the given terms. a5 = a4 r Simplify. –40 = –8r Substitute –40 for a5 and –8 for a4. 5 = r Divide both sides by –8.

Check It Out! Example 3a Continued Step 2 Find a1. an = a1r n - 1 General rule –8 = a1(5)4 - 1 Use a5 = –8 and r = 5. –0.064 = a1

Check It Out! Example 3a Continued Step 3 Write the rule and evaluate for a7. an = a1r n - 1 an = –0.064(5)n - 1 Substitute for a1 and r. a7 = –0.064(5)7 - 1 Evaluate for n = 7. a7 = –1,000 The 7th term is –1,000.

Geometric means are the terms between any two nonconsecutive terms of a geometric sequence.

Example 4: Finding Geometric Means Find the geometric mean of and . Use the formula.

Check It Out! Example 4 Find the geometric mean of 16 and 25. Use the formula.

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 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. = 1(1.96875) ≈ 1.97

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.

Example 6: Sports Application An online video game tournament begins with 1024 players. Four players play in each game, and in each game, only the winner advances to the next round. How many games must be played to determine the winner? Step 1 Write a sequence. Let n = the number of rounds, an = the number of games played in the nth round, and Sn = the total number of games played through n rounds.

Example 6 Continued Step 2 Find the number of rounds required. The final round will have 1 game, so substitute 1 for an. Isolate the exponential expression by dividing by 256. 4 = n – 1 Equate the exponents. 5 = n Solve for n.

Example 6 Continued Step 3 Find the total number of games after 5 rounds. Sum function for geometric series 341 games must be played to determine the winner.

Check It Out! Example 6 A 6-year lease states that the annual rent for an office space is $84,000 the first year and will increase by 8% each additional year of the lease. What will the total rent expense be for the 6-year lease?  $616,218.04

Lesson Quiz: Part I 1. Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference. geometric; r = 6 2. Find the 8th term of the geometric sequence 1, –2, 4, –8, …. –128 3. Find the 9th term of the geometric sequence with a2 = 0.3 and a6 = 0.00003. 0.00000003

Lesson Quiz: Part II 4. Find the geometric mean of and 18. 3 5. Find the indicated sum for the geometric series 40 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? 40