7.3 Analyze Geometric Sequences & Series

What is a geometric sequence? What is the rule for a geometric sequence? How do you find the nth term given 2 terms?

Geometric Sequence The ratio of any term to it’s previous term is constant. This means you multiply by the same number to get each term. This number that you multiply by is called the common ratio (r).

Example: Decide whether each sequence is geometric. 4,-8,16,-32,… -8/4=-2 16/-8=-2 -32/16=-2 Geometric (common ratio is -2) 3, 9, -27,-81, 243,… 9/3 = 3 -27/9= −3 -81/-27= 3 243/-81=−3 Not geometric

Tell whether the sequence is geometric. Explain why or why not. 1. 81, 27, 9, 3, 1, . . . SOLUTION To decide whether a sequence is geometric, find the ratios of consecutive forms. a3 a2 = 9 27 3 1 a2 a1 = 27 81 3 9 1 a4 a3 = 3 9 1 a5 a4 = 1 3 Each ratio is , So the sequence is geometric. 1 3 ANSWER

2. 1, 2, 6, 24, 120, . . . SOLUTION To decide whether a sequence is geometric find the ratios of consecutive terms. a2 a1 = 2 1 a3 a2 = 6 2 3 a4 a3 = 24 6 4 a5 a4 = 120 24 5 ANSWER The ratios are different. The sequence is not geometric.

Rule for a Geometric Sequence   Example: Write a rule for the nth term of the sequence 5, 2, 0.8, 0.32,… . Then find a8. First, find r. r= 2/5 = .4 an=5(.4)n-1 a8=5(.4)8-1 a8=5(.4)7 a8=5(.0016384) a8=.008192

Write a rule for the nth term of the sequence. Then find a7. SOLUTION The sequence is geometric with first term a1 = 4 and common ratio a. r = 20 4 = 5. So, a rule for the nth term is: an = a1 r n – 1 Write general rule. = 4(5)n – 1 Substitute 4 for a1 and 5 for r. The 7th term is a7 = 4(5)7 – 1 = 62,500.

Write a rule for the nth term of the sequence. Then find a7. b. 152, – 76, 38, – 19, . . . SOLUTION The sequence is geometric with first term a1 = 152 and common ratio b. r = –76 152 = – 1 2 .So, a rule for the nth term is: an = a1 r n – 1 Write general rule.   Substitute 152 for a1 and for r. 1 2 –   19 8 =

 

7.3 Assignment, Day 1 Page 454, 4-26 even

Geometric Sequences and Series day 2 How do you find the nth term given 2 terms? What is the formula for finding the sum of an finite geometric series?

Write a rule given a term and the common ratio One term of a geometric sequence is a4 =12. The common ratio is r = 2. a. Write a rule for the nth term. SOLUTION a. Use the general rule to find the first term. an = a1r n – 1 Write general rule. a4 = a1r 4 – 1 Substitute 4 for n. 12 = a1(2)3 Substitute 12 for a4 and 2 for r. 1.5 = a1 Solve for a1. So, a rule for the nth term is: an = a1r n – 1 Write general rule. = 1.5(2) n – 1 Substitute 1.5 for a1 and 2 for r.

Write a rule given a term and the common ratio One term of a geometric sequence is a4 =12. The common ratio is r = 2. b. Graph the sequence. SOLUTION Create a table of values for the sequence. The graph of the first 6 terms of the sequence is shown. Notice that the points lie on an exponential curve. This is true for any geometric sequence with r > 0. b.

Example: One term of a geometric sequence is a4=3 Example: One term of a geometric sequence is a4=3. The common ratio is r=3. Write a rule for the nth term. Then graph the sequence. If a4=3, then when n=4, an=3. Use an=a1rn-1 3=a1(3)4-1 3=a1(3)3 3=a1(27) 1/9=a1 an=a1rn-1 an=(1/9)(3)n-1 To graph, graph the points of the form (n,an). Such as, (1,1/9), (2,1/3), (3,1), (4,3),…

Substitute for a1 in Equation 2. Two terms of a geometric sequence are a3 = 248 and a6 = 3072. Find a rule for the nth term. SOLUTION Write a system of equations using an 5 a1r n – 1 and substituting 3 for n (Equation 1) and then 6 for n (Equation 2). STEP 1 a3 = a1r 3 – 1 – 48 = a1 r 2 Equation 1 Equation 2 a6 = a1r 6 – 1 3072 = a1r 5 STEP 2 Solve the system. – 48 r 2 = a1 Solve Equation 1 for a1. 3072 = – 48 r 2 (r 5 ) Substitute for a1 in Equation 2. 3072 = – 48r 3 Simplify. –4 = r Solve for r. – 48 = a1(– 4)2 Substitute for r in Equation 1. – 3 = a1 Solve for a1.

–4 = r – 3 = a1 STEP 3 an = a1r n – 1 Write general rule. an = – 3(– 4)n – 1 Substitute for a1 and r.

Example: Two terms of a geometric sequence are a2=-4 and a6=-1024 Example: Two terms of a geometric sequence are a2=-4 and a6=-1024. Write a rule for the nth term. Write 2 equations, one for each given term. a2=a1r2-1 OR -4=a1r a6=a1r6-1 OR -1024=a1r5 Use these 2 equations & substitution to solve for a1 & r. -4/r=a1 -1024=(-4/r)r5 -1024=-4r4 256=r4 4=r & -4=r If r=4, then a1=-1. an=(-1)(4)n-1 If r=-4, then a1=1. an=(1)(-4)n-1 an=(-4)n-1 Both Work!

Formula for the Sum of a Finite Geometric Series n = # of terms a1 = 1st term r = common ratio

Example: Consider the geometric series 4+2+1+½+… . Find the sum of the first 10 terms. Find n such that Sn=31/4.

log232=n

Find the sum of the geometric series 4(3)i – 1. 16 i = 1 4(3)i – 1. a1 = 4(3)1– 1 = 4 Identify first term. r = 3 Identify common ratio.   Write rule for S16.   Substitute 4 for a1 and 3 for r. = 86,093,440 Simplify. ANSWER The sum of the series is 86,093,440.

Movie Revenue In 1990, the total box office revenue at U.S. movie theaters was about $5.02 billion. From 1990 through 2003, the total box office revenue increased by about 5.9% per year. Write a rule for the total box office revenue an (in billions of dollars) in terms of the year. Let n = 1 represent 1990. a. SOLUTION Because the total box office revenue increased by the same percent each year, the total revenues from year to year form a geometric sequence. Use a1 = 5.02 and r = 1 + 0.059 = 1.059 to write a rule for the sequence. a. an = 5.02(1.059)n – 1 Write a rule for an.

There are 14 years in the period 1990–2003, so find S14. b. Movie Revenue In 1990, the total box office revenue at U.S. movie theaters was about $5.02 billion. From 1990 through 2003, the total box office revenue increased by about 5.9% per year. What was the total box office revenue at U.S. movie theaters for the entire period 1990–2003? b. There are 14 years in the period 1990–2003, so find S14. b.       ANSWER The total movie box office revenue for the period 1990–2003 was about $105 billion.

7. Find the sum of the geometric series 6( – 2)i–1. SOLUTION a1 = 6( – 2) = 6 Identify first term. r = – 2 Identify common ratio. ( ) 1 – r 8 1 – r = 6 Write rule for S8. 1 – (– 2)8 1 – (– 2) = 6   1 + 2 1 – 16 = 6   = 6 – 255 3   = – 510 ANSWER

How do you find the nth term given 2 terms? Write two equations with two unknowns and solve by substitution. What is the formula for finding the sum of an finite geometric series?

7.3 Assignment, Day 2 p.454 28-52 even