Algebra 2 unit 12.3.12.5

UNIT 12.3/12.5 GEOMETRIC UNIT 12.3/12.5 GEOMETRIC SEQUENCES AND SERIESSEQUENCES AND SERIES

Warm UpSimplify.

1. 2.

3. (–2)8 4.

Solve for x.

5.

Evaluate.

96

256

Find terms of a geometric sequence, including geometric means.

Find the sums of geometric series.

Objectives

geometric sequencegeometric meangeometric series

Vocabulary

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 successiveterms is a constant called the common ratio r (r ≠ 1) . For the players remaining, r is .

Recall that exponential functions have a commonratio. When you graph the ordered pairs (n, an) of ageometric sequence, the points lie on an exponentialcurve as shown. Thus, you can think of a geometricsequence as an exponential function with sequentialnatural numbers as the domain.

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

Differences

It could be geometric with

Ratios

1.7, 1.3, 0.9, 0.5, . . .

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 Differences –0.4 –0.4 –0.4

It could be arithmetic, with r = –0.4.

Ratio

Check It Out! Example 1c

Determine whether each sequence could be geometric or arithmetic. If possible, find the common ratio or difference.

–50, –32, –18, –8, . . . –50, –32, –18, –8, . . .

Differences 18 14 10

It is neither.

Ratios

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 termCommon ratio

First term

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.

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

The 9th term is .

Substitute for a1, 9 for

n, and for r.


Check Extend the sequence.

Given

a6 =

a7 =

a8 =

a9 =

0.001, 0.01, 0.1, 1, 10, . . .

Check It Out! Example 2b

Find the 9th term of the geometric sequence.


Check It Out! Example 2b Continued

Step 2 Write a rule, and evaluate for n = 9.

an = a1 r n–1

a9 = 0.001(10)9–1

= 0.001(100,000,000) = 100,000

The 7th term is 100,000.

General rule

Substitute 0.001 for a1,

9 for n, and 10 for r.


Check Extend the sequence.

a6 = 10(10) = 100

a7 = 100(10) = 1,000

a8 = 1,000(10) = 10,000

a9 = 10,000(10) = 100,000

Givena5 = 10

When given two terms of a sequence, be sure to consider positive and negativevalues for r when necessary.

Caution!


Find the 7th term of the geometric sequence with the given terms.

a4 = –8 and a5 = –40Step 1 Find the common ratio.

a5 = a4 r(5 – 4)

a5 = a4 r

–40 = –8r

5 = r

Use the given terms.

Simplify.

Substitute –40 for a5 and –8 for a4.

Divide both sides by –8.


Step 2 Find a1.

an = a1r n - 1

–8 = a1(5)4 - 1

–0.064 = a1

General rule

Use a5 = –8 and r = 5.


Step 3 Write the rule and evaluate for a7.

an = a1r n - 1

Substitute for a1 and r.

The 7th term is –1,000.

an = –0.064(5)n - 1

a7 = –0.064(5)7 - 1

a7 = –1,000

Evaluate for n = 7.

a2 = 768 and a4 = 48


Find the 7th term of the geometric sequence with the given terms.


a4 = a2 r(4 – 2)

a4 = a2 r2

48 = 768r2

0.0625 = r2

Use the given terms.

Simplify.

Substitute 48 for a4 and 768 for a2.

Divide both sides by 768.

±0.25 = r Take the square root.


Step 2 Find a1.

Consider both the positive and negative values for r.

an = a1r n - 1

768 = a1(0.25)2 - 1

3072 = a1

an = a1r n - 1

768 = a1(–0.25)2 - 1

–3072 = a1

General rule

Use a2= 768 and r = ±0.25.

or


Step 3 Write the rule and evaluate for a7.

Consider both the positive and negative values for r.

an = a1r n - 1 an = a1r n - 1

Substitute for a1 and r.an = 3072(0.25)n - 1

a7 = 3072(0.25)7 - 1

a7 = 0.75

an = 3072(–0.25)n - 1

a7 = 3072(–0.25)7 - 1

a7 = 0.75

Evaluate for n = 7.

or


an = a1r n - 1 an = a1r n - 1

Substitute for a1 and r.

The 7th term is 0.75 or –0.75.

an = –3072(0.25)n - 1

a7 = –3072(0.25)7 - 1

a7 = –0.75

an = –3072(–0.25)n - 1

a7 = –3072(–0.25)7 - 1

a7 = –0.75

Evaluate for n = 7.

or

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

Check It Out! Example 4

Find the geometric mean of 16 and 25.

Use the formula.

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.


Find the indicated sum for each geometric series.


S6 for


Step 2 Find S6 with a1 = 2, r = , and n = 6.

Substitute.

Sum formula


Find the indicated sum for each geometric series.

Step 1 Find the first term.

Step 2 Find S6.


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

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Algebra 2 unit 12.3.12.5