Chapter 11 Sec 3 Geometric Sequences. 2 of 18 Algebra 2 Chapter 11 Sections 3 – 5 Geometric Sequence A geometric sequence is a sequence in which each

Chapter 11 Sec 3Chapter 11 Sec 3

Geometric SequencesGeometric Sequences

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Algebra 2 Chapter 11 Sections 3 – 5

Geometric SequenceGeometric Sequence

• A A geometric sequencegeometric sequence is a sequence in is a sequence in which each term after the first is found by which each term after the first is found by multiplying the previous term by a constant multiplying the previous term by a constant r r called the called the common ratiocommon ratio..

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Example 1Example 1Find the eighth term of a geometric sequence for Find the eighth term of a geometric sequence for

which which aa11 = – 3 and = – 3 and r = r = – 2.– 2.

aann = = aa11 · · r r n – n – 11

aa88 = (–3) · (–2) = (–3) · (–2) 8 – 1 8 – 1

aa88 = (–3) · (–128) = (–3) · (–128)

aa88 = 384 = 384

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Example 2Example 2Write an equation for the n term of a geometric Write an equation for the n term of a geometric

sequence 3, 12, 48, 192…sequence 3, 12, 48, 192…

aann = = aa11 · · rrn – n – 11

aann = (3) · (4) = (3) · (4) n n – 1 – 1

So the equations is So the equations is aann = 3(4) = 3(4) n n – 1 – 1

43

12 and 3

1

21

a

ara

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Example 3Example 3Find the tenth term of a geometric sequence for which Find the tenth term of a geometric sequence for which

aa44 = 108 and = 108 and r r = 3. = 3.

aann = = aa11 · · r r nn – – 11

aa44 = = aa1 1 · (3)· (3) 4 – 1 4 – 1

108 = 108 = aa1 1 · (3)· (3) 3 3

108 = 27108 = 27aa1 1

4 = 4 = aa1 1

aann = = aa11 · · r r nn – – 11

aa1010 = 4 = 4 · (3)· (3) 10 – 1 10 – 1

aa1010 = 4 = 4 · (3)· (3) 9 9

aa1010 = 78,732 = 78,732

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Geometric MeansGeometric MeansAs we saw with arithmetic means, you are given two terms As we saw with arithmetic means, you are given two terms of a geometric sequence and are asked to find the terms of a geometric sequence and are asked to find the terms between, these terms between are called between, these terms between are called geometric means.geometric means.

Find the three geometric means between 3.12 and 49.92.Find the three geometric means between 3.12 and 49.92.

3.12, _____, _____, _____, 49.923.12, _____, _____, _____, 49.92

aann = = aa11 · · r r n – n – 11

aa55 = = 3.12 · 3.12 · r r 55 – – 11

49.92 = 3.12 49.92 = 3.12 r r 44

16 = 16 = r r 44

±2±2 = r So… = r So…

aa1 1 aa22 a a33 a a44 a a55

6.246.24 – – 6.246.24 12.4812.48

24.9624.96––24.9624.96


Geometric SeriesGeometric Series

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Geometric SeriesGeometric SeriesGeometric Sequence Geometric Sequence Geometric Series.Geometric Series.

1, 2, 4, 8, 161, 2, 4, 8, 16 1 + 2 + 4 + 8 + 16 1 + 2 + 4 + 8 + 16

4, –12, 364, –12, 36 4 + (–12) + 36 4 + (–12) + 36

SSnn represents the sum of the first represents the sum of the first nn terms of a series. For terms of a series. For

example, example, SS44 is the sum of the first four terms. is the sum of the first four terms.

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Example 1Example 1

Evaluate Evaluate

6

1

125n

n 525 111 a

166 25 a2

6

r

n

r

raS

n

n

1

11

21

215 6

nS

315

1

635

nS

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Only have the first and last Only have the first and last terms?terms?You can use the formula for finding the nth term in You can use the formula for finding the nth term in

((aann = = aa11 · · r r n – n – 11 ) conjunction with the sum formula) conjunction with the sum formula

when you don’t know when you don’t know n.n.

aann · · r r = = aa11 · · r r n – n – 11 · · rr

aann · · r r = = aa11 · · r r n n

.1

11

r

raaS

n

n

.1

11

r

raaS

n

n

.11

r

raaS n

n

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Example 3Example 3Find Find aa11 in a geometric series for which in a geometric series for which SS8 8 = 39,360= 39,360

and and r = r = 3.3.

r

raS

n

n

1

11

31

31360,39

81

a

2

6560360,39 1

a

1

1

12

3280360,39

a

a


Infinite Geometric Infinite Geometric SeriesSeries

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Infinite Geometric SeriesInfinite Geometric SeriesAny geometric series with an infinite number of terms. Any geometric series with an infinite number of terms.

Consider the infinite geometric series Consider the infinite geometric series

You have already learned to find the sum You have already learned to find the sum SSnn of the first n terms, of the first n terms,

this is called partial sum for an infinite series.this is called partial sum for an infinite series.

...16

1

8

1

4

1

2

1

Notice that as n increases, the partial sum levels off and approaches a limit of one. This leveling-off behavior is characteristic of infinite geometric series for which

| r | < 1.

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Sum of an Infinite SeriesSum of an Infinite SeriesLets use the formula for the sum of a finite series to find a Lets use the formula for the sum of a finite series to find a

formula for an infinite series.formula for an infinite series.

If –1 < If –1 < rr < 1 , the value if < 1 , the value if rrnn will approach 0 as n increases. will approach 0 as n increases.

Therefore the partial sum of the infinite series will approachTherefore the partial sum of the infinite series will approach

r

raaS

n

n

1

11

r

ra

r

aS

n

n

1111

r

a

r

a

r

aSn

1

or 1

0

1111

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Example 1Example 1Find the sum of each infinite geometric series, if it exists.Find the sum of each infinite geometric series, if it exists.

...32

9

8

3

2

1 a. First find the value of First find the value of r r to determine if the sum to determine if the sum

exists.exists.

so 8

3 and

2

121 aa

exists. sum the,14

3 Since ,

4

3

2183

r

r

aSn

1

1

43

1

21

2

4121

...8421 b.

so 2 and 1 21 aaexist.not does

sum the,12 Since ,21

2

r

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Example 2: Sigma Time…Example 2: Sigma Time…

Evaluate Evaluate

1

1

5

124

n

n 241 a5

1r

51

1

24

11

r

aSn 20

5624

205

124

1

1

n

n

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Repeeeeating DecimalRepeeeeating Decimal

Write 0.39 as a fraction.Write 0.39 as a fraction.

S = 0.39S = 0.39

S = 0 .393939393939… thenS = 0 .393939393939… then

100S = 39.393939393939… Subtract 100S – S100S = 39.393939393939… Subtract 100S – S

– – S = 0 .393939393939… S = 0 .393939393939…

99S = 3999S = 39

typo typo intentionalintentional

33

13

99

39S

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Daily AssignmentDaily Assignment

• Chapter 11 Sections 3 – 5 Chapter 11 Sections 3 – 5 • Study Guide (SG)Study Guide (SG)

• Pg 145 – 150 OddPg 145 – 150 Odd

Documents

Chapter 11 Sec 3 Geometric Sequences. 2 of 18 Algebra 2 Chapter 11 Sections 3 – 5 Geometric Sequence A geometric sequence is a sequence in which each