slide 1 / 145 slide 2 / 145 - center for teaching &...
TRANSCRIPT
Slide 1 / 145 Slide 2 / 145
Pre-Calculus
Sequences and Series
www.njctl.org
2015-03-24
Slide 3 / 145
Table of Contents
Arithmetic Sequences
Arithmetic Series
Geometric Sequences
Geometric Series
Special Sequences
Infinite Geometric Series
Binomial Theorem
click on the topic to go to that section
Slide 4 / 145
Arithmetic Sequences
Return to Table of Contents
Slide 5 / 145 Slide 6 / 145
Slide 7 / 145
1 Find the next term in the arithmetic sequence: 3, 9, 15, 21, . . .
Arithmetic Sequence
Teac
her
Teac
her
Slide 8 / 145
2 Find the next term in the arithmetic sequence: -8, -4, 0, 4, . . .
Arithmetic Sequence
Teac
her
Teac
her
Slide 9 / 145
3 Find the next term in the arithmetic sequence: 2.3, 4.5, 6.7, 8.9, . . .
Arithmetic SequenceTe
ache
rTe
ache
r
Slide 10 / 145
4 Find the value of d in the arithmetic sequence: 10, -2, -14, -26, . . .
Arithmetic Sequence
Teac
her
Teac
her
Slide 11 / 145
5 Find the value of d in the arithmetic sequence: -8, 3, 14, 25, . . .
Arithmetic Sequence
Teac
her
Teac
her
Slide 12 / 145
Arithmetic SequenceAs we study sequences we need a way of naming the terms. a1 to represent the first term,a2 to represent the second term,a3 to represent the third term,and so on in this manner.
If we were talking about the 8th term we would use a8.
When we want to talk about general term call it the nth termand use an.
Arithmetic Sequence
Slide 13 / 145
Arithmetic SequenceWrite the first four terms of the arithmetic sequence that is described.
a1 = 4; d = 6
a1 = 3; d = -3
a1 = 0.5; d = 2.3
a2 = 7; d = 5
Arithmetic Sequence
Teac
her
Teac
her
Slide 14 / 145
6 Which sequence matches the description?
A 4, 6, 8, 10
B 2, 6,10, 14
C 2, 8, 32, 128
D 4, 8, 16, 32
Arithmetic Sequence
Teac
her
Teac
her
Slide 15 / 145
7 Which sequence matches the description?
A -3, -7, -10, -14
B -4, -7, -11, -13
C -3, -7, -11, -15
D -3, 1, 5, 9
Arithmetic SequenceTe
ache
rTe
ache
r
Slide 16 / 145
8 Which sequence matches the description?
A 7, 10, 13, 16
B 4, 7, 10, 13
C 1, 4, 7,10
D 3, 5, 7, 9
Arithmetic Sequence
Teac
her
Teac
her
Slide 17 / 145
Arithmetic Sequence
To find a specific term,say the 5th or a5, you couldwrite out all of the terms.
But what about the 100th term(or a100 )?
We need to find a formula to get there directly without writing out the whole list.
Arithmetic Sequence
Slide 18 / 145
Arithmetic Sequence
Consider: 3, 9, 15, 21, 27, 33, 39,. . .
a1 3
a2 9 = 3+6
a3 15 = 3+12 = 3+2(6)
a4 21 = 3+18 = 3+3(6)
a5 27 = 3+24 = 3+ 4(6)
a6 33 = 3+30 = 3+5(6)
a7 39 = 3+36 = 3+6(6)
Do you see a pattern that relates the term number to its
value?
Arithmetic Sequence
Teac
her
Teac
her
click
Slide 19 / 145
Arithmetic SequenceExample Find the 21st term of the arithmetic sequence with a1 = 4 and d = 3.
Example Find the 12th term of the arithmetic sequence with a1 = 6 and d = -5.
Arithmetic Sequence
Teac
her
Teac
her
Slide 20 / 145
Arithmetic SequenceExample Find the 1st term of the arithmetic sequence with a15 = 30 and d = 7.
Example Find the 1st term of the arithmetic sequence with a17 = 4 and d = -2.
Arithmetic Sequence
Teac
her
Teac
her
Slide 21 / 145
Arithmetic SequenceExample Find d of the arithmetic sequence with a15 = 42 and a1=3.
Example Find the term number n of the arithmetic sequence with an = 6, a1=-34 and d = 4.
Arithmetic SequenceTe
ache
rTe
ache
r
Slide 22 / 145
9 Find a11 when a1 = 13 and d = 6.
Arithmetic Sequence
Teac
her
Teac
her
Slide 23 / 145
10 Find a17 when a1 = 12 and d = -0.5
Arithmetic Sequence
Teac
her
Teac
her
Slide 24 / 145
11 Find a17 for the sequence 2, 4.5, 7, 9.5, ...
Arithmetic Sequence
Teac
her
Teac
her
Slide 25 / 145
12 Find the common difference d when a1 = 12 and a13= 6.
Arithmetic Sequence
Teac
her
Teac
her
Slide 26 / 145
13 Find n such a1 = 12 , an= -20, and d = -2.
Arithmetic Sequence
Teac
her
Teac
her
Slide 27 / 145
14 Tom works at a car dealership selling cars. He is paid $4000 a month plus a $300 commission for every car he sells after the first car . How much did he make in April if he sold 14 cars?
Arithmetic SequenceTe
ache
rTe
ache
r
Slide 28 / 145
Arithmetic Sequence
Find the missing terms in the arithmetic sequence.
4, 6, 8, 10,___
5, 10, ___, 20
___, 12, 9, 6
6,___, 14
Notice in the last example d was added to get from 6 to ___ and another d was added to get from ___ to 14.
Or 6 + 2d = 14
Arithmetic Sequence
Teac
her
Teac
her
Slide 29 / 145
Arithmetic Sequence
Find the missing terms
2, ___ , ___ , 23
4, ___ , ___ , -14
7, ___ , ___, ___, 39
-9, ___ , ___ , ___, ___ , -34
Arithmetic Sequence
Teac
her
Teac
her
Slide 30 / 145
15 Find the missing term: 4, ___ , -16
A -20
B -10
C -6
D 2
Arithmetic Sequence
Teac
her
Teac
her
Slide 31 / 145
16 Find the missing terms: -10, ___ , ___, 8
A -6, -2
B -6, 2
C -5, 1
D -4, 2
Arithmetic Sequence
Teac
her
Teac
her
Slide 32 / 145
17 Find the missing terms: 12, ___ , ___, 75
A 27, 51
B 33, 54
C 37, 51
D 34, 58
Arithmetic Sequence
Teac
her
Teac
her
Slide 33 / 145
18 Find d for the arithmetic sequence: 5, ___ , ___ , ___ , 21
Arithmetic SequenceTe
ache
rTe
ache
r
Slide 34 / 145
Arithmetic Series
Return to Table of Contents
Slide 35 / 145
Arithmetic SeriesAn arithmetic series is the sum of the terms in the arithmetic sequence.
Sn represents the sum of the first n terms.
Consider 4, 7,10, 13, 16, 19, 22, . . .
S4= 4 + 7 + 10 + 13 = 34
S6= 4 + 7 + 10 + 13 + 16 + 19= 69
Arithmetic Series
Slide 36 / 145
19 Consider 3, 9, 15, 21, 27, 33, 39,... what is S4?
Arithmetic Series
Teac
her
Teac
her
Slide 37 / 145
20 Consider 3, 9, 15, 21, 27, 33, 39,... what is S5?
Arithmetic Series
Teac
her
Teac
her
Slide 38 / 145
21 Consider 3, 9, 15, 21, 27, 33, 39,... what is S7?
Arithmetic Series
Teac
her
Teac
her
Slide 39 / 145
Arithmetic SeriesSuppose we wanted to find the first 100 terms of 4, 7,10, 13, 16, 19, 22, . . . or S 100?
There must be a short cut.
The 100th term of the sequence is 301 using a100 = 4 + (100 - 1)3.
S100 = 4 + 7 + 10 +13 + . . . + 292 + 295 +298 + 301If we add the smallest and largest (4 + 301)= 305If we add the next two (7 + 298) = 305and continue (10 + 295) until they are all paired up (151+154).
We now have 50 pairs of 305, so...S100 = 4 + 7 + 10 +. . .+298 + 301 = 50(305) = 15250
Do you see a pattern?
Arithmetic Series
Slide 40 / 145
Slide 41 / 145
Find Sn for each arithmetic series.
Example: a1 = 7 and a12 = -23
Example: a1 = 6, n = 10, and d = 9
Arithmetic Series
Teac
her
Teac
her
Slide 42 / 145
Find Sn for each arithmetic series.Example: a12= 30 and d = -7
Example: a1 = 2, an = 32, and d = 5
Arithmetic Series
Teac
her
Teac
her
Slide 43 / 145
22 Find the Sn for the arithmetic series described:a1 = 19 and a12 = 37.
Arithmetic Series
Teac
her
Teac
her
Slide 44 / 145
23 Find the Sn for the arithmetic series described:a1 = 30 and a17 = -45.
Arithmetic Series
Teac
her
Teac
her
Slide 45 / 145
24 Find the Sn for the arithmetic series described:a1 = 20, n = 8, and d = 6.
Arithmetic SeriesTe
ache
rTe
ache
r
Slide 46 / 145
25 Find the Sn for the arithmetic series described:an = 20, n = 9, and d = -4.
Arithmetic Series
Teac
her
Teac
her
Slide 47 / 145
26 Find the Sn for the arithmetic series described:17 + 20 + 23 + 26 + 29 + . . . + 50
Arithmetic Series
Teac
her
Teac
her
Slide 48 / 145
Sigma Notation
Sigma ( ) is the Greek letter S.
And means the sum of the terms in a sequence.
The difference between Sn and sigma is that Sn is always the first to the nth term.
means start with 3rd and sum up through the 9th term.
Arithmetic Series
Slide 49 / 145 Slide 50 / 145
27 Evaluate
Arithmetic Series
Teac
her
Teac
her
Slide 51 / 145
28 Evaluate
Arithmetic SeriesTe
ache
rTe
ache
r
Slide 52 / 145
29 Evaluate
Arithmetic Series
Teac
her
Teac
her
Slide 53 / 145
How would evaluate ?
Arithmetic Series
Teac
her
Teac
her
Slide 54 / 145
30 Evaluate
Arithmetic Series
Teac
her
Teac
her
Slide 55 / 145
Geometric Sequences
Return to Table of Contents
Slide 56 / 145
Slide 57 / 145
31 Find the next term in geometric sequence: 6, -12, 24, -48, 96, . . .
Geometric SequenceTe
ache
rTe
ache
r
Slide 58 / 145
32 Find the next term in geometric sequence: 64, 16, 4, 1, . . .
Geometric Sequence
Teac
her
Teac
her
Slide 59 / 145
33 Find the next term in geometric sequence: 6, 15, 37.5, 93.75, . . .
Geometric Sequence
Teac
her
Teac
her
Slide 60 / 145
34 Is the following sequence a geomtric one? 48, 24, 12, 8, 4, 2, 1
Yes
No
Geometric Sequence
Teac
her
Teac
her
Slide 61 / 145
Geometric SequenceGeometric Sequences can be described by giving the
first term, a1, and the common ratio, r.
Examples: Find the first five terms of the geometric sequence described.
1) a1 = 6 and r = 3
2) a1 = 8 and r = -.5
3) a1 = -24 and r = 1.5
4) a1 = 12 and r = 2/3
Geometric Sequence
Teac
her
Teac
her
Slide 62 / 145
35 Find the first four terms of the geometric sequence described: a1 = 6 and r = 4.
A 6, 24, 96, 384
B 4, 24, 144, 864
C 6, 10, 14, 18
D 4, 10, 16, 22
Geometric Sequence
Teac
her
Teac
her
Slide 63 / 145
36 Find the first four terms of the geometric sequence described: a1 = 12 and r = -1/2.
A 12, -6, 3, -.75
B 12, -6, 3, -1.5
C 6, -3, 1.5, -.75
D -6, 3, -1.5, .75
Geometric Sequence
Teac
her
Teac
her
Slide 64 / 145
37 Find the first four terms of the geometric sequence described: a1 = 7 and r = -2.
A 14, 28, 56, 112
B -14, 28, -56, 112
C 7, -14, 28, -56
D -7, 14, -28, 56
Geometric Sequence
Teac
her
Teac
her
Slide 65 / 145
Geometric SequenceConsider the sequence: 3, 6, 12, 24, 48, 96, . . .To find the seventh term, just multiply the sixth term by 2.But what if I want to find the 20 th term?
Look for a pattern:a1 3
a2 6 = 3(2)
a3 12 = 3(4) = 3(2)2
a4 24 = 3(8) = 3(2)3
a5 48 = 3(16) = 3(2)4
a6 96 = 3(32) = 3(2)5
a7 192 = 3(64) = 3(2)6
Do you see a pattern?
Geometric Sequence
click
Slide 66 / 145
Geometric Sequence
Geometric Sequence
Slide 67 / 145
Geometric SequenceFind the indicated term.
Example: a20 given a1 =3 and r = 2.
Example: a10 for 2187, 729, 243, 81
Geometric Sequence
Teac
her
Teac
her
Slide 68 / 145
Geometric SequenceExample: Find r if a6 = .2 and a1 = 625
Example: Find n if a1 = 6, an = 98,304 and r = 4.
Geometric Sequence
Teac
her
Teac
her
Slide 69 / 145
38 Find a12 in a geometric sequence wherea1 = 5 and r = 3.
Geometric SequenceTe
ache
rTe
ache
r
Slide 70 / 145
39 Find a10 in a geometric sequence wherea1 = 7 and r = -2.
Geometric Sequence
Teac
her
Teac
her
Slide 71 / 145
40 Find a7 in a geometric sequence wherea1 = 10 and r = -1/2.
Geometric Sequence
Teac
her
Teac
her
Slide 72 / 145
41 Find r of a geometric sequence wherea1 = 3 and a10=59049.
Geometric Sequence
Teac
her
Teac
her
Slide 73 / 145
42 Find n of a geometric sequence wherea1 = 72, r = .5, and an = 2.25
Geometric Sequence
Teac
her
Teac
her
Slide 74 / 145
Geometric SequenceFind the missing term in the geometric sequence3, 9, 27,___
5, 1, 1/5 , ____
___, -10, 50, -250
-2, ___, -32
Geometric Sequence
Teac
her
Teac
her
Slide 75 / 145
Geometric Sequence
Find the missing terms in the geometric sequence.
5, ___, ___, 40
-54, ___, ___, 16
4, ___, ___, ___, 324
144, ___, ___, ___, ___, 4.5
Geometric SequenceTe
ache
rTe
ache
r
Slide 76 / 145
43 What number(s) fill in the blanks of the geometric sequence: ___, 14, 98, 686
A -14
B -7
C -2
D 2
E 7
F 8
G 10
H 12
I 28
J 50
Geometric Sequence
Teac
her
Teac
her
Slide 77 / 145 Slide 78 / 145
Slide 79 / 145
Geometric Series
Return to Table of Contents
Slide 80 / 145
The sum of a geometric series can be found using the formula:
Examples: Find Sn for each example.
1) a1= 5, r= 3, n= 6
2) a1= -3, r= -2, n=7
Geometric Series
Teac
her
Teac
her
Slide 81 / 145
Sometimes information will be missing, so that using
isn't possible to start. Look to use to find missing information.Example: a1 = 16 and a5 = 243, find S5
Geometric SeriesTe
ache
rTe
ache
r
Slide 82 / 145
Geometric Series
Teac
her
Teac
her
Example: a1 = 16 and a5 = 243, find S5 (continued)
Slide 83 / 145
46 Find the indicated sum of the geometric series described: a1 = 10, n = 6, and r = 6
Geometric Series
Teac
her
Teac
her
Slide 84 / 145
47 Find the indicated sum of the geometric series described: a1 = -2, n = 5, and r = 1/4
Geometric Series
Teac
her
Teac
her
Slide 85 / 145
48 Find the indicated sum of the geometric series described: a1 = 8, n = 6, and r = -2
Geometric Series
Teac
her
Teac
her
Slide 86 / 145
49 Find the indicated sum of the geometric series described: a1 = 8, n = 5, and a6 = 8192
Geometric Series
Teac
her
Teac
her
Slide 87 / 145
50 Find the indicated sum of the geometric series described: r = 6, n = 4, and a4 = 2592
Geometric SeriesTe
ache
rTe
ache
r
Slide 88 / 145
51 Find the indicated sum of the geometric series described: 8 - 12 + 18 - . . . find S7
Geometric Series
Teac
her
Teac
her
Slide 89 / 145
Sigma ( )can be used to describe the sum of a geometric series.
Examples:
We can still use , but to do so we must examine the sigma notation.
n = 4 Why? The bounds on below and on top indicate that.
a1 = 6 Why? The coefficient is all that remains when the base is powered by 0.
r = 3 Why? In the exponential chapter this was our growth rate.
Geometric Series
Teac
her
Teac
her
Slide 90 / 145
52 Find the sum:
Geometric Series
Teac
her
Teac
her
Slide 91 / 145
53 Find the sum:
Geometric Series
Teac
her
Teac
her
Slide 92 / 145
54 Find the sum:
Geometric Series
Teac
her
Teac
her
Slide 93 / 145
InfiniteGeometric Series
Return to Table of Contents
Slide 94 / 145
n Sn
1 3
2 12
3 39
4 120
5 363
6 1,092
7 3,279
8 9,840
n an
1 3
2 9
3 27
4 81
5 243
6 729
7 2,187
8 6,561Because r = 3, the series isgrowing at increasing rate.
Infinite Geometric Series
Slide 95 / 145
n Sn
1 64
2 96
3 112
4 120
5 124
6 126
7 127
8 127.5
n an
1 64
2 32
3 16
4 8
5 4
6 2
7 1
8 0.5 Because r = 1/2 , an --> 0.Notice Sn --> an asymptote?
Infinite Geometric Series
Slide 96 / 145
n Sn
1 64
2 32
3 48
4 40
5 44
6 42
7 43
8 42.5
n an
1 64
2 -32
3 16
4 -8
5 4
6 -2
7 1
8 -0.5 Because r = -1/2 , an --> 0.Notice Sn --> an asymptote?
Infinite Geometric Series
Slide 97 / 145
For an infinite geometric series to approach a value,-1 < r < 1 then
The examples from the previous slides:
Example 1: a1 = 3 and r = 3.
Example 2: a1 = 64 and r= 1/2
Example 3: a1 = 64 and r= -1/2
Infinite Geometric Series
Teac
her
Teac
her
Slide 98 / 145
55 Find the sum of this infinite geometric series, if one exists:
Infinite Geometric Series
Teac
her
Teac
her
Slide 99 / 145 Slide 100 / 145
Slide 101 / 145 Slide 102 / 145
59 Find the sum of this infinite geometric series, if one exists:
Infinite Geometric Series
Teac
her
Teac
her
Slide 103 / 145
60 Find the sum of this infinite geometric series, if one exists:
Infinite Geometric Series
Teac
her
Teac
her
Slide 104 / 145
Special Sequences
Return to Table of Contents
Slide 105 / 145
A recursive formula is one in which to find a term you need to know the preceding term.
So to know term 8 you need the value of term 7, and to know the nth term you need term n-1
In each example, find the first 5 terms
a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an-1 +3
6 a1 10 a1 12
13 a2 40 a2 27
20 a3 160 a3 57
27 a4 640 a4 117
34 a5 2560 a5 237
Special Sequences
Slide 106 / 145
61 Find the first four terms of the sequence:
A 6, 3, 0, -3
B 6, -18, 54, -162
C -3, 3, 9, 15
D -3, 18, 108, 648
a1 = 6 and an = an-1 - 3
Special Sequences
Teac
her
Teac
her
Slide 107 / 145
62 Find the first four terms of the sequence:
A 6, 3, 0, -3
B 6, -18, 54, -162
C -3, 3, 9, 15
D -3, 18, 108, 648
a1 = 6 and an = -3an-1
Special Sequences
Teac
her
Teac
her
Slide 108 / 145
63 Find the first four terms of the sequence:
A 6, -22, 70, -216
B 6, -22, 70, -214
C 6, -14, 46, -134
D 6, -14, 46, -142
a1 = 6 and an = -3an-1 + 4
Special Sequences
Teac
her
Teac
her
Slide 109 / 145
a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an-1 +3
6 a1 10 a1 12
13 a2 40 a2 27
20 a3 160 a3 57
27 a4 640 a4 117
34 a5 2560 a5 237
The recursive formula in the first column represents an Arithmetic Sequence.We can write this formula so that we find a n directly.
Recall:
We will need a1 and d,they can be found both from the table and the recursive formula.
Special Sequences
Teac
her
Teac
her
Slide 110 / 145
a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an-1 +3
6 a1 10 a1 12
13 a2 40 a2 27
20 a3 160 a3 57
27 a4 640 a4 117
34 a5 2560 a5 237
The recursive formula in the second column represents a Geometric Sequence.We can write this formula so that we find an directly.
Recall:
We will need a1 and r,they can be found both from the table and the recursive formula.
Special Sequences
Teac
her
Teac
her
Slide 111 / 145
a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an-1 +3
6 a1 10 a1 12
13 a2 40 a2 27
20 a3 160 a3 57
27 a4 640 a4 117
34 a5 2560 a5 237
The recursive formula in the third column represents neither an Arithmetic or Geometric Sequence.
This observation comes from the formula where you have both multiply and add from one term to the next.
Special Sequences
Slide 112 / 145
64 Identify the sequence as arithmetic, geometric,or neither.
A Arithmetic
B Geometric
C Neither
a1 = 12 , an = 2an-1 +7
Special Sequences
Teac
her
Teac
her
Slide 113 / 145
65 Identify the sequence as arithmetic, geometric,or neither.
A Arithmetic
B Geometric
C Neither
a1 = 20 , an = 5an-1
Special Sequences
Teac
her
Teac
her
Slide 114 / 145
66 Which equation could be used to find the nth term of the recursive formula directly?
A
B
C
D
a1 = 20 , an = 5an-1
an = 20 + (n-1)5
an = 20(5)n-1
an = 5 + (n-1)20
an = 5(20)n-1
Special Sequences
Teac
her
Teac
her
Slide 115 / 145
67 Identify the sequence as arithmetic, geometric,or neither.
A Arithmetic
B Geometric
C Neither
a1 = -12 , an = an-1 - 8
Special Sequences
Teac
her
Teac
her
Slide 116 / 145
68 Which equation could be used to find the nth term of the recursive formula directly?
A
B
C
D
an = -12 + (n-1)(-8)
an = -12(-8)n-1
an = -8 + (n-1)(-12)
an = -8(-12)n-1
a1 = -12 , an = an-1 - 8
Special Sequences
Teac
her
Teac
her
Slide 117 / 145
69 Identify the sequence as arithmetic, geometric,or neither.
A Arithmetic
B Geometric
C Neither
a1 = 10 , an = an-1 + 8
Special SequencesTe
ache
rTe
ache
r
Slide 118 / 145
70 Which equation could be used to find the nth term of the recursive formula directly?
A
B
C
D
an = 10 + (n-1)(8)
an = 10(8)n-1
an = 8 + (n-1)(10)
an = 8(10)n-1
a1 = 10 , an = an-1 + 8
Special Sequences
Teac
her
Teac
her
Slide 119 / 145
71 Identify the sequence as arithmetic, geometric,or neither.
A Arithmetic
B Geometric
C Neither
a1 = 24 , an = (1/2)an-1
Special Sequences
Teac
her
Teac
her
Slide 120 / 145
72 Which equation could be used to find the nth term of the recursive formula directly?
A
B
C
D
an = 24 + (n-1)(1/2)
an = 24(1/2)n-1
an = (1/2) + (n-1)24
an = (1/2)(24)n-1
a1 = 24 , an = (1/2)an-1
Special Sequences
Teac
her
Teac
her
Slide 121 / 145
Special Recursive Sequences
Some recursive sequences not only rely on the preceding term, but on the two preceding terms.
4
7
7 + 4 = 11
11 + 7 = 18
18 + 11 =29
Find the first five terms of the sequence:a1 = 4, a2 = 7, and an = an-1 + an-2
Special Sequences
Slide 122 / 145
Special Recursive Sequences
Some recursive sequences not only rely on the preceding term, but on the two preceding terms.
6
8
2(8) + 3(6) = 34
2(34) + 3(8) = 92
2(92) + 3(34) = 286
Find the first five terms of the sequence:a1 = 6, a2 = 8, and an = 2an-1 + 3an-2
Special Sequences
Slide 123 / 145
Special Recursive Sequences
Some recursive sequences not only rely on the preceding term, but on the two preceding terms.
10
6
2(6) -10 = 2
2(2) - 6 = -2
2(-2) - 2= -6
Find the first five terms of the sequence:a1 = 10, a2 = 6, and an = 2an-1 - an-2
Special Sequences
Slide 124 / 145
Special Recursive Sequences
Some recursive sequences not only rely on the preceding term, but on the two preceding terms.
1
1
1+1 = 2
1 + 2 = 3
2 + 3 =5
Find the first five terms of the sequence:a1 = 1, a2 = 1, and an = an-1 + an-2
Special Sequences
Slide 125 / 145
The sequence in the preceding example is called
The Fibonacci Sequence1, 1, 2, 3, 5, 8, 13, 21, . . .
where the first 2 terms are 1'sand any term there after is the sum of preceding two terms.
This is as famous as a sequence can get and is worth remembering.
Special Sequences
Slide 126 / 145
73 Find the first four terms of sequence:
A
B
C
D
a1 = 5, a2 = 7, and an = a1 + a2
7, 5, 12, 19
5, 7, 35, 165
5, 7, 12, 19
5, 7, 13, 20
Special Sequences
Teac
her
Teac
her
Slide 127 / 145
74 Find the first four terms of sequence:
A
B
C
D
a1 = 4, a2 = 12, and an = 2an-1 - an-2
4, 12, -4, -20
4, 12, 4, 12
4, 12, 20, 28
4, 12, 20, 36
Special Sequences
Teac
her
Teac
her
Slide 128 / 145
75 Find the first four terms of sequence:
A
B
C
D
a1 = 3, a2 = 3, and an = 3an-1 + an-2
3, 3, 6, 9
3, 3, 12, 39
3, 3, 12, 36
3, 3, 6, 21
Special Sequences
Teac
her
Teac
her
Slide 129 / 145
Binomial Theorem
Return to Table of Contents
Slide 130 / 145
Look for a pattern when powering a binomial.
Binomial Theorem
Slide 131 / 145
One pattern comes from the coefficients.
11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 11 7 21 35 35 21 7 1
This is Pascal's Triangle.
Binomial Theorem
Slide 132 / 145
Slide 133 / 145
Example: Find the coefficient of the 4 term of 6th power of a binomial.
Binomial Theorem
Teac
her
Teac
her
Slide 134 / 145
76 Calculate
Binomial Theorem
Teac
her
Teac
her
Slide 135 / 145
77 Calculate
Binomial TheoremTe
ache
rTe
ache
r
Slide 136 / 145
78 Calculate
Binomial Theorem
Teac
her
Teac
her
Slide 137 / 145
79 What is the binomial coefficient of the 4th term of x + y to the 8th power?
Binomial Theorem
Teac
her
Teac
her
Slide 138 / 145
80 What is the binomial coefficient of the 2nd term of x + y to the 3rd power?
Binomial Theorem
Teac
her
Teac
her
Slide 139 / 145
The second pattern of powering a binomial comes from the exponents.
Click to see the Binomial Expansion.
But what if instead of x + y we have 2x -3?
Binomial Theorem
Slide 140 / 145
The Binomial Theorem
Binomial Theorem
Slide 141 / 145
Example: Expand (2x -3)4
Binomial TheoremTe
ache
rTe
ache
r
Slide 142 / 145
Slide 143 / 145
81 What is the coefficient of the 4th term (2x - 3) to the 5th power?
Binomial Theorem
Teac
her
Teac
her
Slide 144 / 145
82 What is the coefficient of the 3rd term (2x - 3) to the 6th power?
Binomial Theorem
Teac
her
Teac
her
Slide 145 / 145
83 What is the coefficient of the 6th term (2x - 3) to the 7th power?
Binomial Theorem
Teac
her
Teac
her