algebra 2 geometric sequences and series powerpoint 2012

BellworkSimplify.

1. 2.

3. (–2)8 4.

Solve for x.

5.

Evaluate.

96

256

Geometric Sequences and Series

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.

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

Example 1A: Identifying Geometric Sequences

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

180, 90, 60, 15, ...

180, 90, 60, 15

Differences –90 –30 –45

It is neither.

3Ratios 1 1 1

2 4


Example 1C: Identifying Geometric Sequences

5, 1, 0.2, 0.04, ...

5, 1, 0.2, 0.04

Differences –4 –0.8 –0.16

5Ratios 1 1 1

5 5

It could be geometric, with

Try 1a


Differences

It could be geometric with

Ratios

1.7, 1.3, 0.9, 0.5, . . .

Try 1b


1.7 1.3 0.9 0.5

Differences –0.4 –0.4 –0.4

It could be arithmetic, with r = –0.4.

Ratio

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

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.

Find the 7th term of the geometric sequence 3, 12, 48, 192, ....

Example 2: Finding the nth Term Given a Geometric Sequence

Step 1 Find the common ratio.

r = a2

a1

123

= 4=

Example 2 Continued

Step 2 Write a rule, and evaluate for n = 7.

an = a1 r n–1

a7 = 3(4)7–1

= 3(4096) = 12,288

The 7th term is 12,288.

General rule

Substitute 3 for a1,7 for n, and 4 for r.

Try 2a

Find the 9th term of the geometric sequence.


Try 2a Continued


an = a1 r n–1 General rule

The 9th term is .

Substitute for a1, 9 for

n, and for r.

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

Try 2b

Find the 9th term of the geometric sequence.


Try 2b Continued


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.

Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324.

Example 3: Finding the nth Term Given Two Terms


a5 = a3 r(5 – 3)

a5 = a3 r2

324 = 36r2

9 = r2

3 = r

Use the given terms.

Simplify.

Substitute 324 for a5 and 36 for a3.

Divide both sides by 36.

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

36 = a1(3)3 - 1

4 = a1

an = a1r n - 1

36 = a1(–3)3 - 1

4 = a1

General rule

Use a3 = 36 and r = 3.

or

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

Substitute a1 and r.

The 8th term is 8748 or –8747.

an = 4(3)n - 1

a8 = 4(3)8 - 1

a8 = 8748

an = 4(–3)n - 1

a8 = 4(–3)8 - 1

a8 = –8748

Evaluate for n = 8.

General rule

or

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

Caution!

Try 3a

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.

Try 3a Continued

Step 2 Find a1.

an = a1r n - 1

–8 = a1(5)4 - 1

–0.064 = a1

General rule

Use a5 = –8 and r = 5.

Try 3a Continued

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.

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

Example 4: Finding Geometric Means

Use the formula.

Find the geometric mean of and .

Try 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 the geometric series.

Example 5A: Finding the Sum of a Geometric Series

S8 for 1 + 2 + 4 + 8 + 16 + ...


Example 5A Continued

Step 2 Find S8 with a1 = 1, r = 2, and n = 8.

Check Use a graphing calculator.

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.

= 1(1.96875) ≈ 1.97

Check Use a graphing calculator.

Try 5a

Find the indicated sum for each geometric series.


S6 for

Try 5a Continued

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

Substitute.

Sum formula

algebra 2 geometric sequences and series powerpoint 2012

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