Copyright © 2007 Pearson Education, Inc. Slide 11-1 DONE BY, AFSAL M NAHAS XI B KV PATTOM
Nov 15, 2014
Copyright © 2007 Pearson Education, Inc. Slide 11-1
DONE BY,
AFSAL M NAHAS XI B KV PATTOM
Copyright © 2007 Pearson Education, Inc. Slide 11-2
SEQUENCE AND SERIES
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An introduction…………
1, 4, 7,10,13
9,1, 7, 15
6.2, 6.6, 7, 7.4
, 3, 6
Arithmetic Sequences
ADDTo get next term
2, 4, 8,16, 32
9, 3,1, 1/ 3
1,1/ 4,1/16,1/ 64
, 2.5 , 6.25
Geometric Sequences
MULTIPLYTo get next term
Arithmetic Series
Sum of Terms
35
12
27.2
3 9
Geometric Series
Sum of Terms
62
20 / 3
85 / 64
9.75
Copyright © 2007 Pearson Education, Inc. Slide 11-9
Find the next four terms of –9, -2, 5, …
Arithmetic Sequence
2 9 5 2 7
7 is referred to as the common difference (d)
Common Difference (d) – what we ADD to get next term
Next four terms……12, 19, 26, 33
Copyright © 2007 Pearson Education, Inc. Slide 11-10
Find the next four terms of 0, 7, 14, …
Arithmetic Sequence, d = 7
21, 28, 35, 42
Find the next four terms of x, 2x, 3x, …
Arithmetic Sequence, d = x
4x, 5x, 6x, 7x
Find the next four terms of 5k, -k, -7k, …
Arithmetic Sequence, d = -6k
-13k, -19k, -25k, -32k
Copyright © 2007 Pearson Education, Inc. Slide 11-11
Vocabulary of Sequences (Universal)
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
n 1
n 1 n
nth term of arithmetic sequence
sum of n terms of arithmetic sequen
a a n 1 d
nS a a
2ce
Copyright © 2007 Pearson Education, Inc. Slide 11-12
Given an arithmetic sequence with 15 1a 38 and d 3, find a .
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
x
15
38
NA
-3
n 1a a n 1 d
38 x 1 15 3
X = 80
Copyright © 2007 Pearson Education, Inc. Slide 11-13
63Find S of 19, 13, 7,...
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
-19
63
??
x
6
n 1a a n 1 d
?? 19 6 1
?? 353
3 6
353
n 1 n
nS a a
2
63
633 3S
219 5
63 1 1S 052
Copyright © 2007 Pearson Education, Inc. Slide 11-14
16 1Find a if a 1.5 and d 0.5 Try this one:
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
1.5
16
x
NA
0.5
n 1a a n 1 d
16 1.5 0.a 16 51
16a 9
Copyright © 2007 Pearson Education, Inc. Slide 11-15
n 1Find n if a 633, a 9, and d 24
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
9
x
633
NA
24
n 1a a n 1 d
633 9 21x 4
633 9 2 244x
X = 27
Copyright © 2007 Pearson Education, Inc. Slide 11-16
1 29Find d if a 6 and a 20
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
-6
29
20
NA
x
n 1a a n 1 d
120 6 29 x
26 28x
13x
14
Copyright © 2007 Pearson Education, Inc. Slide 11-17
Find two arithmetic means between –4 and 5
-4, ____, ____, 5
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
-4
4
5
NA
x
n 1a a n 1 d
15 4 4 x x 3
The two arithmetic means are –1 and 2, since –4, -1, 2, 5
forms an arithmetic sequence
Copyright © 2007 Pearson Education, Inc. Slide 11-18
Find three arithmetic means between 1 and 4
1, ____, ____, ____, 4
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
1
5
4
NA
x
n 1a a n 1 d
4 1 x15 3
x4
The three arithmetic means are 7/4, 10/4, and 13/4
since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence
Copyright © 2007 Pearson Education, Inc. Slide 11-19
Find n for the series in which 1 na 5, d 3, S 440
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
5
x
y
440
3
n 1a a n 1 d
n 1 n
nS a a
2
y 5 31x
x40 y4
25
12
x440 5 5 x 3
x 7 x440
2
3
880 x 7 3x 20 3x 7x 880
X = 16
Graph on positive window
Copyright © 2007 Pearson Education, Inc. Slide 11-20
Copyright © 2007 Pearson Education, Inc. Slide 11-21Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21
An infinite sequence is a function whose domain is the set of positive integers.
a1, a2, a3, a4, . . . , an, . . .
The first three terms of the sequence an = 2n2 are
a1 = 2(1)2 = 2
a2 = 2(2)2 = 8
a3 = 2(3)2 = 18.
finite sequence
terms
Copyright © 2007 Pearson Education, Inc. Slide 11-22Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22
A sequence is geometric if the ratios of consecutive terms are the same.
2, 8, 32, 128, 512, . . .
geometric sequence
The common ratio, r, is 4.
82
4
328
4
12832
4
512128
4
Copyright © 2007 Pearson Education, Inc. Slide 11-23Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23
The nth term of a geometric sequence has the form
an = a1rn - 1
where r is the common ratio of consecutive terms of the sequence.
15, 75, 375, 1875, . . . a1 = 15
The nth term is 15(5n-1).
75 515
r
a2 = 15(5)
a3 = 15(52)
a4 = 15(53)
Copyright © 2007 Pearson Education, Inc. Slide 11-24Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24
Example: Find the 9th term of the geometric sequence
7, 21, 63, . . .
a1 = 7
The 9th term is 45,927.
21 37
r
an = a1rn – 1 = 7(3)n – 1
a9 = 7(3)9 – 1 = 7(3)8
= 7(6561) = 45,927
Copyright © 2007 Pearson Education, Inc. Slide 11-25Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25
The sum of the first n terms of a sequence is represented by summation notation.
1 2 3 41
n
i ni
a a a a a a
index of summation
upper limit of summation
lower limit of summation
5
1
4n
n
1 2 3 4 54 4 4 4 4 4 16 64 256 1024 1364
Copyright © 2007 Pearson Education, Inc. Slide 11-26Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26
The sum of a finite geometric sequence is given by
11 1
1
1 .1
n nin
i
rS a r ar
5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ?
n = 8
a1 = 5
1
81 11
221
5n
nrS ar
5210r
1 25651 2 2555
1 1275
Copyright © 2007 Pearson Education, Inc. Slide 11-27Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27
The sum of the terms of an infinite geometric sequence is called a geometric series.
a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . .
If |r| < 1, then the infinite geometric series
11
0
.1
i
i
aS a r
r
has the sum
If 1 , then the series does not have a sum.r
Copyright © 2007 Pearson Education, Inc. Slide 11-28Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28
Example: Find the sum of
1
1a
Sr
1 13 13 9
13
r
3
1 13
3 31 413 3
The sum of the series is 9 .4
3 934 4
Copyright © 2007 Pearson Education, Inc. Slide 11-29
11.3 Geometric Sequences and Series
1, 2, 4, 8, 16 … is an example of a geometric sequence with first term 1 and each subsequent term is 2 times the term preceding it.
The multiplier from each term to the next is called the common ratio and is usually denoted by r.
A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number.
Copyright © 2007 Pearson Education, Inc. Slide 11-30
11.3 Finding the Common Ratio
In a geometric sequence, the common ratio can be found by dividing any term by the term preceding it.
The geometric sequence 2, 8, 32, 128, …has common ratio r = 4 since
8 32 128... 4
2 8 32
Copyright © 2007 Pearson Education, Inc. Slide 11-31
11.3 Geometric Sequences and Series
nth Term of a Geometric Sequence
In the geometric sequence with first term a1 and common ratio r, the nth term an, is
11
nna a r
Copyright © 2007 Pearson Education, Inc. Slide 11-32
11.3 Using the Formula for the nth Term
Example Find a5 and an for the geometric
sequence 4, –12, 36, –108 , …
Solution Here a1= 4 and r = 36/ –12 = – 3. Using
n=5 in the formula
In general
5 1 45 4 ( 3) 4 ( 3) 324a
1 11 4 ( 3)n n
na a r
11
nna a r
Copyright © 2007 Pearson Education, Inc. Slide 11-33
11.3 Modeling a Population of Fruit Flies
Example A population of fruit flies grows in such a way that each generation is 1.5 times the previous generation. There were 100 insects in the first generation. How many are in the fourth generation.
Solution The populations form a geometric sequence with a1= 100 and r = 1.5 . Using n=4 in the formula
for an gives
or about 338 insects in the fourth generation.
3 34 1 100(1.5) 337.5a a r
Copyright © 2007 Pearson Education, Inc. Slide 11-34
11.3 Geometric Series
A geometric series is the sum of the terms of a geometric sequence .
In the fruit fly population model with a1 = 100 and r = 1.5, the total population after four generations is a geometric series:
1 2 3 4
2 3100 100(1.5) 100(1.5) 100(1.5)
813
a a a a
Copyright © 2007 Pearson Education, Inc. Slide 11-35
11.3 Geometric Sequences and Series
Sum of the First n Terms of an Geometric Sequence
If a geometric sequence has first term a1 and common ratio r, then the sum of the first n terms is given by
where .1(1 )
1
n
n
a rS
r
1r
Copyright © 2007 Pearson Education, Inc. Slide 11-36
11.3 Finding the Sum of the First n Terms
Example Find
Solution This is the sum of the first six terms of a
geometric series with and r = 3.
From the formula for Sn ,
.
11 2 3 6a
6
1
2 3i
i
6
6
6(1 3 ) 6(1 729) 6( 728)2184
1 3 2 2S
Copyright © 2007 Pearson Education, Inc. Slide 11-37
Vocabulary of Sequences (Universal)
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
n 1
n 1
n1
n
nth term of geometric sequence
sum of n terms of geometric sequ
a a r
a r 1S
r 1ence
Copyright © 2007 Pearson Education, Inc. Slide 11-38
Find the next three terms of 2, 3, 9/2, ___, ___, ___
3 – 2 vs. 9/2 – 3… not arithmetic3 9 / 2 3
1.5 geometric r2 3 2
3 3 3 3 3 3
2 2 2
92, 3, , , ,
2
9 9 9
2 2 2 2 2 2
92, 3, , ,
27 81 243
4 8,
2 16
Copyright © 2007 Pearson Education, Inc. Slide 11-39
1 9
1 2If a , r , find a .
2 3
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
1/2
x
9
NA
2/3
n 1n 1a a r
9 11 2
x2 3
8
8
2x
2 3
7
8
2
3 128
6561
Copyright © 2007 Pearson Education, Inc. Slide 11-40
Find two geometric means between –2 and 54
-2, ____, ____, 54
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
-2
54
4
NA
x
n 1n 1a a r
1454 2 x
327 x 3 x
The two geometric means are 6 and -18, since –2, 6, -18, 54
forms an geometric sequence
Copyright © 2007 Pearson Education, Inc. Slide 11-41
2 4 1
2Find a a if a 3 and r
3
-3, ____, ____, ____
2Since r ...
3
4 83, 2, ,
3 9
2 4
8 10a a 2
9 9
Copyright © 2007 Pearson Education, Inc. Slide 11-42
9Find a of 2, 2, 2 2,...
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
x
9
NA
2
2 2 2r 2
22
n 1n 1a a r
9 1
x 2 2
8
x 2 2
x 16 2
Copyright © 2007 Pearson Education, Inc. Slide 11-43
5 2If a 32 2 and r 2, find a
____, , ____,________ ,32 2
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
x
5
NA
32 2
2n 1
n 1a a r
5 1
32 2 x 2
4
32 2 x 2
32 2 x4
8 2 x
Copyright © 2007 Pearson Education, Inc. Slide 11-44
*** Insert one geometric mean between ¼ and 4***
*** denotes trick question
1,____,4
4
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
1/4
3
NA
4
xn 1
n 1a a r
3 114
4r 2r
14
4 216 r 4 r
1,1, 4
4
1, 1, 4
4
Copyright © 2007 Pearson Education, Inc. Slide 11-45
7
1 1 1Find S of ...
2 4 8
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
1/2
7
x
NA
11184r
1 1 22 4
n1
n
a r 1S
r 1
71 12 2
x12
1
1
71 12 2
12
1
63
64
Copyright © 2007 Pearson Education, Inc. Slide 11-46
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Copyright © 2007 Pearson Education, Inc. Slide 11-50
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Copyright © 2007 Pearson Education, Inc. Slide 11-52
1, 4, 7, 10, 13, …. Infinite Arithmetic No Sum
3, 7, 11, …, 51 Finite Arithmetic n 1 n
nS a a
2
1, 2, 4, …, 64 Finite Geometric n
1
n
a r 1S
r 1
1, 2, 4, 8, … Infinite Geometricr > 1r < -1
No Sum
1 1 13,1, , , ...
3 9 27Infinite Geometric
-1 < r < 11a
S1 r
Copyright © 2007 Pearson Education, Inc. Slide 11-53
Find the sum, if possible: 1 1 1
1 ...2 4 8
1 112 4r
11 22
1 r 1 Yes
1a 1S 2
11 r 12
Copyright © 2007 Pearson Education, Inc. Slide 11-54
Find the sum, if possible: 2 2 8 16 2 ...
8 16 2r 2 2
82 2 1 r 1 No
NO SUM
Copyright © 2007 Pearson Education, Inc. Slide 11-55
Find the sum, if possible: 2 1 1 1
...3 3 6 12
1 113 6r
2 1 23 3
1 r 1 Yes
1
2a 43S
11 r 312
Copyright © 2007 Pearson Education, Inc. Slide 11-56
Find the sum, if possible: 2 4 8
...7 7 7
4 87 7r 22 47 7
1 r 1 No
NO SUM
Copyright © 2007 Pearson Education, Inc. Slide 11-57
Find the sum, if possible: 5
10 5 ...2
55 12r
10 5 2 1 r 1 Yes
1a 10S 20
11 r 12
Copyright © 2007 Pearson Education, Inc. Slide 11-58
The Bouncing Ball Problem – Version A
A ball is dropped from a height of 50 feet. It rebounds 4/5 of
it’s height, and continues this pattern until it stops. How far
does the ball travel?50
40
32
32/5
40
32
32/5
40S 45
504
10
1554
Copyright © 2007 Pearson Education, Inc. Slide 11-59
The Bouncing Ball Problem – Version B
A ball is thrown 100 feet into the air. It rebounds 3/4 of
it’s height, and continues this pattern until it stops. How far
does the ball travel?
100
75
225/4
100
75
225/4
10S 80
100
4 43
1
0
10
3
Copyright © 2007 Pearson Education, Inc. Slide 11-60
11.3 Infinite Geometric Series
If a1, a2, a3, … is a geometric sequence and the sequence of sums S1, S2, S3, …is a convergent
sequence, converging to a number S. Then S is
said to be the sum of the infinite geometric series 1 2 3 ...a a a S
Copyright © 2007 Pearson Education, Inc. Slide 11-61
11.3 An Infinite Geometric Series
Given the infinite geometric sequence
the sequence of sums is S1 = 2, S2 = 3, S3 = 3.5, …
1 1 1 12, 1, , , , ,...
2 4 8 16
The calculator screen shows more sums, approaching a value of 4. So
1 12 1 ... 4
2 4
Copyright © 2007 Pearson Education, Inc. Slide 11-62
11.3 Infinite Geometric Series
Sum of the Terms of an Infinite Geometric Sequence
The sum of the terms of an infinite geometric sequence with first term a1 and common ratio r, where –1 < r < 1 is given by
.1
1
aS
r
Copyright © 2007 Pearson Education, Inc. Slide 11-63
11.3 Finding Sums of the Terms of Infinite Geometric Sequences
Example Find
Solution Here and so
.
1
3
5
i
i
1
3
5a
1
1
33 35
35 1 215
i
i
a
r
3
5r
Copyright © 2007 Pearson Education, Inc. Slide 11-64