Try It
The first five terms are { 1,6, 11, 16, 21 }. { 1,6, 11, 16, 21 }.
The first five terms are { −2, 2, − 3 2 , 1,− 5 8 }. { −2, 2, − 3 2 , 1,− 5 8 }.
The first six terms are { 2,5,54,10,250,15 }. { 2,5,54,10,250,15 }.
a n = (−1) n+1 9 n a n = (−1) n+1 9 n
a n =− 3 n 4n a n =− 3 n 4n
a n = e n−3 a n = e n−3
{ 2, 5, 11, 23, 47 } { 2, 5, 11, 23, 47 }
{ 0, 1, 1, 1, 2, 3, 5 2 , 17 6 }. { 0, 1, 1, 1, 2, 3, 5 2 , 17 6 }.
The first five terms are { 1, 3 2 , 4,15,72 }. { 1, 3 2 , 4,15,72 }.
The sequence is arithmetic. The common difference is –2. –2.
The sequence is not arithmetic because 3−1≠6−3. 3−1≠6−3.
{ 1 , 6 , 11 , 16 , 21 } { 1 , 6 , 11 , 16 , 21 }
a 2 = 2 a 2 = 2
a 1 = 25 a n = a n − 1 + 12 , for n ≥ 2 a 1 = 25 a n = a n − 1 + 12 , for n ≥ 2
a n = 53 − 3 n a n = 53 − 3 n
There are 11 terms in the sequence.
The formula is T n =10+4n, T n =10+4n, and it will take her 42 minutes.
The sequence is not geometric because 10 5 ≠ 15 10 10 5 ≠ 15 10 .
The sequence is geometric. The common ratio is 1 5 1 5 .
{ 18,6,2, 2 3 , 2 9 } { 18,6,2, 2 3 , 2 9 }
a 1 =2 a n = 2 3 a n−1 for n≥2 a 1 =2 a n = 2 3 a n−1 for n≥2
a 6 =16,384 a 6 =16,384
a n =− (−3) n−1 a n =− (−3) n−1
1. 293⋅1.026<sup>n</sup>
2. ਹਿੱਟਾਂ ਦੀ ਗਿਣਤੀ ਲਗਭਗ 333 ਹੋਵੇਗੀ।
3. 38
4. 26.4 26.4
5. 328 328
6. −280 −280
7. $2,025
8. ≈2,000.00 ≈2,000.00
9. 9,840
10. $275,513.31
11. ਜੋੜ ਪਰਿਭਾਸ਼ਿਤ ਨਹੀਂ ਹੈ।
12. ਅਨੰਤ ਲੜੀ ਦਾ ਜੋੜ ਪਰਿਭਾਸ਼ਿਤ ਹੈ।
13. ਅਨੰਤ ਲੜੀ ਦਾ ਜੋੜ ਪਰਿਭਾਸ਼ਿਤ ਹੈ।
14. 3
15. ਲੜੀ ਜਿਓਮੈਟ੍ਰਿਕ ਨਹੀਂ ਹੈ।
16. − 3 11 − 3 11
17. $32,775.87
18. 7
19. 60 ਸੰਭਵ ਨਾਸ਼ਤੇ ਦੇ ਵਿਸ਼ੇਸ਼ ਹਨ।
20. 120
21. 60
22. 12
23. P(7,7)=5,040 P(7,7)=5,040
24. P(7,5)=2,520 P(7,5)=2,520
C(10,3)=120 C(10,3)=120
64 sundaes
840
ⓐ35
ⓑ330
ⓐ x 5 −5 x 4 y+10 x 3 y 2 −10 x 2 y 3 +5x y 4 − y 5 x 5 −5 x 4 y+10 x 3 y 2 −10 x 2 y 3 +5x y 4 − y 5
ⓑ 8 x 3 +60 x 2 y+150x y 2 +125 y 3 8 x 3 +60 x 2 y+150x y 2 +125 y 3
−10,206 x 4 y 5 −10,206 x 4 y 5
row: Outcome | Probability
row: Heads | 1212
row: Tails | 1212
2 3 2 3
7 13 7 13
2 13 2 13
5 6 5 6
a. 1 91 ; b. 5 91 ; c. 86 91 a. 1 91 ; b. 5 91 ; c. 86 91
13.1 Section Exercises
A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers.
Yes, both sets go on indefinitely, so they are both infinite sequences.
A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out 13⋅12⋅11⋅10⋅9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1. 13⋅12⋅11⋅10⋅9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1.
First four terms: −8,− 16 3 ,−4,− 16 5 −8,− 16 3 ,−4,− 16 5
First four terms: 2, 1 2 , 8 27 , 1 4 2, 1 2 , 8 27 , 1 4 .
First four terms: 1.25,−5,20,−80 1.25,−5,20,−80 .
First four terms: 1 3 , 4 5 , 9 7 , 16 9 1 3 , 4 5 , 9 7 , 16 9 .
First four terms: − 4 5 ,4,−20,100 − 4 5 ,4,−20,100
1 3 , 4 5 , 9 7 , 16 9 , 25 11 ,31,44,59 1 3 , 4 5 , 9 7 , 16 9 , 25 11 ,31,44,59
−0.6,−3,−15,−20,−375,−80,−9375,−320 −0.6,−3,−15,−20,−375,−80,−9375,−320
a n = n 2 +3 a n = n 2 +3
a n = 2 n 2n or 2 n−1 n a n = 2 n 2n or 2 n−1 n
a n = ( − 1 2 ) n−1 a n = ( − 1 2 ) n−1
First five terms: 3,−9,27,−81,243 3,−9,27,−81,243
First five terms: −1,1,−9, 27 11 , 891 5 −1,1,−9, 27 11 , 891 5
1 24 ,1, 1 4 , 3 2 , 9 4 , 81 4 , 2187 8 , 531,441 16 1 24 ,1, 1 4 , 3 2 , 9 4 , 81 4 , 2187 8 , 531,441 16
2,10,12, 14 5 , 4 5 ,2,10,12 2,10,12, 14 5 , 4 5 ,2,10,12
a 1 =−8, a n = a n−1 +n a 1 =−8, a n = a n−1 +n
a 1 =35, a n = a n−1 +3 a 1 =35, a n = a n−1 +3
720 720
665,280 665,280
First four terms: 1, 1 2 , 2 3 , 3 2 1, 1 2 , 2 3 , 3 2
First four terms: −1,2, 6 5 , 24 11 −1,2, 6 5 , 24 11
a n = 2 n−2 a n = 2 n−2
a 1 =6, a n =2 a n−1 −5 a 1 =6, a n =2 a n−1 −5
First five terms: 29 37 29 37, 152 111 152 111, 716 333 716 333, 3188 999 3188 999, 13724 2997 13724 2997
First five terms: 2, 3, 5, 17, 65537
a 10 =7,257,600 a 10 =7,257,600
First six terms: 0.042, 0.146, 0.875, 2.385, 4.708
First four terms: 5.975, 2.765, 185.743, 1057.25, 6023.521
If a n =−421 a n =−421 is a term in the sequence, then solving the equation −421=−6−8n −421=−6−8n for n n will yield a non-negative integer. However, if −421=−6−8n, −421=−6−8n, then n=51.875 n=51.875 so a n =−421 a n =−421 is not a term in the sequence.
a 1 =1, a 2 =0, a n = a n−1 − a n−2 a 1 =1, a 2 =0, a n = a n−1 − a n−2
(n+2)! (n−1)! = (n+2)·(n+1)·(n)·(n−1)·...·3·2·1 (n−1)·...·3·2·1 =n(n+1)(n+2)= n 3 +3 n 2 +2n (n+2)! (n−1)! = (n+2)·(n+1)·(n)·(n−1)·...·3·2·1 (n−1)·...·3·2·1 =n(n+1)(n+2)= n 3 +3 n 2 +2n
13.2 Section Exercises
A sequence where each successive term of the sequence increases (or decreases) by a constant value.
We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.
Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.
The common difference is 1 2 1 2
The sequence is not arithmetic because 16−4≠64−16. 16−4≠64−16.
0, 2 3 , 4 3 ,2, 8 3 0, 2 3 , 4 3 ,2, 8 3
0 , − 5 , − 10 , − 15 , − 20 0 , − 5 , − 10 , − 15 , − 20
a 4 =19 a 4 =19
a 6 =41 a 6 =41
a 1 =2 a 1 =2
a 1 =5 a 1 =5
a 1 =6 a 1 =6
a 21 =−13.5 a 21 =−13.5
−19,−20.4,−21.8,−23.2,−24.6 −19,−20.4,−21.8,−23.2,−24.6
a 1 =17; a n = a n−1 +9 n≥2 a 1 =17; a n = a n−1 +9 n≥2
a 1 =12; a n = a n−1 +5 n≥2 a 1 =12; a n = a n−1 +5 n≥2
a 1 =8.9; a n = a n−1 +1.4 n≥2 a 1 =8.9; a n = a n−1 +1.4 n≥2
a 1 = 1 5 ; a n = a n−1 + 1 4 n≥2 a 1 = 1 5 ; a n = a n−1 + 1 4 n≥2
1 = 1 6 ; a n = a n−1 − 13 12 n≥2 1 = 1 6 ; a n = a n−1 − 13 12 n≥2
a 1 =4; a n = a n−1 +7; a 14 =95 a 1 =4; a n = a n−1 +7; a 14 =95
First five terms: 20,16,12,8,4. 20,16,12,8,4.
a n =1+2n
a n =−105+100n
a n =1.8n
a n =13.1+2.7n
a n = 1 3 n− 1 3
ਲੜੀ ਵਿੱਚ 10 ਪਦ ਹਨ।
ਲੜੀ ਵਿੱਚ 6 ਪਦ ਹਨ।
ਗ੍ਰਾਫ਼ ਅੰਕਗਣਿਤਕ ਲੜੀ ਨੂੰ ਨਹੀਂ ਦਰਸਾਉਂਦਾ ਹੈ।
1,4,7,10,13,16,19
ਉੱਤਰ ਵੱਖੋ-ਵੱਖਰੇ ਹੋਣਗੇ। ਉਦਾਹਰਨਾਂ: a n =20.6n ਅਤੇ a n =2+20.4n.
a 11 =−17a+38b
13ਵੇਂ ਪਦ 'ਤੇ ਲੜੀ ਨਕਾਰਾਤਮਕ ਮੁੱਲ ਪ੍ਰਾਪਤ ਕਰਨ ਲੱਗ ਪੈਂਦੀ ਹੈ, a 13 =− 1 3
ਉੱਤਰ ਵੱਖੋ-ਵੱਖਰੇ ਹੋਣਗੇ। ਜਾਂਚ ਕਰੋ ਕਿ ਲੜੀ ਅੰਕਗਣਿਤਕ ਹੈ। ਉਦਾਹਰਨ: ਰੈਕਰਸਿਵ ਫਾਰਮੂਲਾ: a 1 =3, a n = a n−1 −3. ਪਹਿਲੇ 4 ਪਦ: 3,0,−3,−6 a 31 =−87
13.3 ਭਾਗ ਅਭਿਆਸ
ਇੱਕ ਲੜੀ ਜਿਸ ਵਿੱਚ ਕਿਸੇ ਵੀ ਦੋ ਲਗਾਤਾਰ ਪਦਾਂ ਦੇ ਅਨੁਪਾਤ ਸਥਿਰ ਹੁੰਦਾ ਹੈ।
ਇੱਕ ਲੜੀ ਦੇ ਹਰੇਕ ਪਦ ਨੂੰ ਪਿਛਲੇ ਪਦ ਨਾਲ ਭਾਗ ਦਿਓ। ਜੇਕਰ ਨਤੀਜੇ ਵਜੋਂ ਆਉਣ ਵਾਲੇ ਭਾਗਫਲ ਬਰਾਬਰ ਹਨ, ਤਾਂ ਲੜੀ ਗੁਣਨਖੰਡੀ ਹੈ।
ਗੁਣਨਖੰਡੀ ਲੜੀਆਂ ਅਤੇ ਘਾਤੀ ਫੰਕਸ਼ਨਾਂ ਦੋਵਾਂ ਵਿੱਚ ਇੱਕ ਸਥਿਰ ਅਨੁਪਾਤ ਹੁੰਦਾ ਹੈ। ਹਾਲਾਂਕਿ, ਉਹਨਾਂ ਦੇ ਡੋਮੇਨ ਇੱਕੋ ਜਿਹੇ ਨਹੀਂ ਹੁੰਦੇ। ਘਾਤੀ ਫੰਕਸ਼ਨ ਸਾਰੇ ਅਸਲ ਨੰਬਰਾਂ ਲਈ ਪਰਿਭਾਸ਼ਿਤ ਕੀਤੇ ਗਏ ਹਨ, ਅਤੇ ਗੁਣਨਖੰਡੀ ਲੜੀਆਂ ਸਿਰਫ ਧਨ ਪੂਰਨ ਅੰਕਾਂ ਲਈ ਪਰਿਭਾਸ਼ਿਤ ਕੀਤੀਆਂ ਗਈਆਂ ਹਨ। ਇੱਕ ਹੋਰ ਅੰਤਰ ਇਹ ਹੈ ਕਿ ਗੁਣਨਖੰਡੀ ਲੜੀ ਦਾ ਅਧਾਰ (ਸਾਂਝਾ ਅਨੁਪਾਤ) ਨਕਾਰਾਤਮਕ ਹੋ ਸਕਦਾ ਹੈ, ਪਰ ਘਾਤੀ ਫੰਕਸ਼ਨ ਦਾ ਅਧਾਰ ਧਨ ਹੋਣਾ ਚਾਹੀਦਾ ਹੈ।
ਸਾਂਝਾ ਅਨੁਪਾਤ −2 ਹੈ
ਲੜੀ ਗੁਣਨਖੰਡੀ ਹੈ। ਸਾਂਝਾ ਅਨੁਪਾਤ 2 ਹੈ।
ਲੜੀ ਗੁਣਨਖੰਡੀ ਹੈ। ਸਾਂਝਾ ਅਨੁਪਾਤ − 1 2 ਹੈ।
ਲੜੀ ਗੁਣਨਖੰਡੀ ਹੈ। ਸਾਂਝਾ ਅਨੁਪਾਤ 5 ਹੈ।
5,1, 1 5 , 1 25 , 1 125
800,400,200,100,50
a 4 =− 16 27
a 7 =− 2 729 a 7 =− 2 729
7,1.4,0.28,0.056,0.0112 7,1.4,0.28,0.056,0.0112
a = 1 −32, a n = 1 2 a n−1 a = 1 −32, a n = 1 2 a n−1
a 1 =10, a n =−0.3 a n−1 a 1 =10, a n =−0.3 a n−1
a 1 = 3 5 , a n = 1 6 a n−1 a 1 = 3 5 , a n = 1 6 a n−1
a 1 = 1 512 , a n =−4 a n−1 a 1 = 1 512 , a n =−4 a n−1
12,−6,3,− 3 2 , 3 4 12,−6,3,− 3 2 , 3 4
a n = 3 n−1 a n = 3 n−1
a n =0.8⋅ (−5) n−1 a n =0.8⋅ (−5) n−1
a n =− ( 4 5 ) n−1 a n =− ( 4 5 ) n−1
a n =3⋅ ( − 1 3 ) n−1 a n =3⋅ ( − 1 3 ) n−1
a 12 = 1 177,147 a 12 = 1 177,147
There are 12 12 terms in the sequence.
The graph does not represent a geometric sequence.
Answers will vary. Examples: a 1 =800, a n =0.5a n−1 a 1 =800, a n =0.5a n−1 and a 1 =12.5, a n =4a n−1 a 1 =12.5, a n =4a n−1
a 5 =256b a 5 =256b
The sequence exceeds 100 100 at the 14th term, a 14 ≈107. a 14 ≈107.
a 4 =− 32 3 a 4 =− 32 3 is the first non-integer value
Answers will vary. Example: Explicit formula with a decimal common ratio: a n =400⋅ 0.5 n−1 ; a n =400⋅ 0.5 n−1 ; First 4 terms: 400,200,100,50; a 8 =3.125 400,200,100,50; a 8 =3.125
13.4 Section Exercises
An nth nth partial sum is the sum of the first n n terms of a sequence.
A geometric series is the sum of the terms in a geometric sequence.
An annuity is a series of regular equal payments that earn a constant compounded interest.
∑ n=0 4 5n ∑ n=0 4 5n
∑ k=1 5 4 ∑ k=1 5 4
∑ k=1 20 8k+2 ∑ k=1 20 8k+2
S 5 = 5( 3 2 + 7 2 ) 2 S 5 = 5( 3 2 + 7 2 ) 2
S 13 = 13( 3.2+5.6 ) 2 S 13 = 13( 3.2+5.6 ) 2
∑ k=1 7 8⋅ 0.5 k−1 ∑ k=1 7 8⋅ 0.5 k−1
S 5 = 9( 1− ( 1 3 ) 5 ) 1− 1 3 = 121 9 ≈13.44 S 5 = 9( 1− ( 1 3 ) 5 ) 1− 1 3 = 121 9 ≈13.44
S 11 = 64( 1− 0.2 11 ) 1−0.2 = 781,249,984 9,765,625 ≈80 S 11 = 64( 1− 0.2 11 ) 1−0.2 = 781,249,984 9,765,625 ≈80
The series is defined. S= 2 1−0.8 S= 2 1−0.8
The series is defined. S= −1 1−( − 1 2 ) S= −1 1−( − 1 2 )
Sample answer: The graph of S n S n seems to be approaching 1. This makes sense because ∑ k=1 ∞ ( 1 2 ) k ∑ k=1 ∞ ( 1 2 ) k is a defined infinite geometric series with S= 1 2 1–( 1 2 ) =1. S= 1 2 1–( 1 2 ) =1.
49
254
S 7 = 147 2 S 7 = 147 2
S 11 = 55 2 S 11 = 55 2
S 7 =5208.4 S 7 =5208.4
S 10 =− 1023 256 S 10 =− 1023 256
S=− 4 3 S=− 4 3
S=9.2 S=9.2
$3,705.42
$695,823.97
a k =30−k a k =30−k
9 terms
r= 4 5 r= 4 5
$400 per month
420 feet
12 feet
13.5 Section Exercises
There are m+n m+n ways for either event A A or event B B to occur.
The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem.
A combination; C(n,r)= n! (n−r)!r! C(n,r)= n! (n−r)!r!
4+2=6 4+2=6
5+4+7=16 5+4+7=16
2×6=12 2×6=12
10 3 =1000 10 3 =1000
P(5,2)=20 P(5,2)=20
P(3,3)=6 P(3,3)=6
P(11,5)=55,440 P(11,5)=55,440
C(12,4)=495 C(12,4)=495
C(7,6)=7 C(7,6)=7
2 10 =1024 2 10 =1024
2 12 =4096 2 12 =4096
2 9 =512 2 9 =512
8! 3! =6720 8! 3! =6720
12! 3!2!3!4! 12! 3!2!3!4!
9
Yes, for the trivial cases r=0 r=0 and r=1. r=1. If r=0, r=0, then C(n,r)=P(n,r)=1. C(n,r)=P(n,r)=1. If r=1, r=1, then r=1, r=1, C(n,r)=P(n,r)=n. C(n,r)=P(n,r)=n.
6! 2! ×4!=8640 6! 2! ×4!=8640
6−3+8−3=8 6−3+8−3=8
4×2×5=40 4×2×5=40
4×12×3=144 4×12×3=144
P(15,9)=1,816,214,400 P(15,9)=1,816,214,400
C(10,3)×C(6,5)×C(5,2)=7,200 C(10,3)×C(6,5)×C(5,2)=7,200
2 11 =2048 2 11 =2048
20! 6!6!8! =116,396,280 20! 6!6!8! =116,396,280
13.6 Section Exercises
A binomial coefficient is an alternative way of denoting the combination C(n,r). C(n,r). It is defined as ( n r )=C(n,r)= n! r!(n−r)! . ( n r )=C(n,r)= n! r!(n−r)! .
The Binomial Theorem is defined as (x+y) n = ∑ k=0 n ( n k ) x n−k y k (x+y) n = ∑ k=0 n ( n k ) x n−k y k and can be used to expand any binomial.
15
35
10
12,376
64 a 3 −48 a 2 b+12a b 2 − b 3 64 a 3 −48 a 2 b+12a b 2 − b 3
27 a 3 +54 a 2 b+36a b 2 +8 b 3 27 a 3 +54 a 2 b+36a b 2 +8 b 3
1024 x 5 +2560 x 4 y+2560 x 3 y 2 +1280 x 2 y 3 +320x y 4 +32 y 5 1024 x 5 +2560 x 4 y+2560 x 3 y 2 +1280 x 2 y 3 +320x y 4 +32 y 5
1024 x 5 −3840 x 4 y+5760 x 3 y 2 −4320 x 2 y 3 +1620x y 4 −243 y 5 1024 x 5 −3840 x 4 y+5760 x 3 y 2 −4320 x 2 y 3 +1620x y 4 −243 y 5
1 x 4 + 8 x 3 y + 24 x 2 y 2 + 32 x y 3 + 16 y 4 1 x 4 + 8 x 3 y + 24 x 2 y 2 + 32 x y 3 + 16 y 4
a 17 +17 a 16 b+136 a 15 b 2 a 17 +17 a 16 b+136 a 15 b 2
a 15 −30 a 14 b+420 a 13 b 2 a 15 −30 a 14 b+420 a 13 b 2
3,486,784,401 a 20 +23,245,229,340 a 19 b+73,609,892,910 a 18 b 2 3,486,784,401 a 20 +23,245,229,340 a 19 b+73,609,892,910 a 18 b 2
x 24 −8 x 21 y +28 x 18 y x 24 −8 x 21 y +28 x 18 y
−720 x 2 y 3 −720 x 2 y 3
220,812,466,875,000 y 7 220,812,466,875,000 y 7
35 x 3 y 4 35 x 3 y 4
1,082,565 a 3 b 16 1,082,565 a 3 b 16
1152 y 2 x 7 1152 y 2 x 7
f 2 (x)= x 4 +12 x 3 f 2 (x)= x 4 +12 x 3
f 4 (x)= x 4 +12 x 3 +54 x 2 +108x f 4 (x)= x 4 +12 x 3 +54 x 2 +108x
590,625 x 5 y 2 590,625 x 5 y 2
k−1 k−1
The expression ( x 3 +2 y 2 −z) 5 ( x 3 +2 y 2 −z) 5 cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.
13.7 Section Exercises
probability; The probability of an event is restricted to values between 0 0 and 1, 1, inclusive of 0 0 and 1. 1.
An experiment is an activity with an observable result.
The probability of the union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets A and B A and B and a union of events A and B, A and B, the union includes either A or B A or B or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is always a numerical value between 0 0 and 1. 1.
1 2 . 1 2 .
5 8 . 5 8 .
1 2 . 1 2 .
3 8 . 3 8 .
1 4 . 1 4 .
3 4 . 3 4 .
3 8 . 3 8 .
1 8 . 1 8 .
15 16 . 15 16 .
5 8 . 5 8 .
1 13 . 1 13 .
1 26 . 1 26 .
12 13 . 12 13 .
row: 1 | 2 | 3 | 4 | 5 | 6
row: 1 | (1,1)2 | (1,2)3 | (1,3)4 | (1,4)5 | (1,5)6 | (1,6)7
row: 2 | (2,1)3 | (2,2)4 | (2,3)5 | (2,4)6 | (2,5)7 | (2,6)8
row: 3 | (3,1)4 | (3,2)5 | (3,3)6 | (3,4)7 | (3,5)8 | (3,6)9
row: 4 | (4,1)5 | (4,2)6 | (4,3)7 | (4,4)8 | (4,5)9 | (4,6)10
row: 5 | (5,1)6 | (5,2)7 | (5,3)8 | (5,4)9 | (5,5)10 | (5,6)11
row: 6 | (6,1)7 | (6,2)8 | (6,3)9 | (6,4)10 | (6,5)11 | (6,6)12
5 12 . 5 12 .
0. 0.
4 9 . 4 9 .
1 4 . 1 4 .
5 8 5 8
8 13 8 13
C(12,5) C(48,5) = 1 2162 C(12,5) C(48,5) = 1 2162
C(12,3)C(36,2) C(48,5) = 175 2162 C(12,3)C(36,2) C(48,5) = 175 2162
C(20,3)C(60,17) C(80,20) ≈12.49% C(20,3)C(60,17) C(80,20) ≈12.49%
C(20,5)C(60,15) C(80,20) ≈23.33% C(20,5)C(60,15) C(80,20) ≈23.33%
20.50+23.33−12.49=31.34% 20.50+23.33−12.49=31.34%
C(40000000,1)C(277000000,4) C(317000000,5) =36.78% C(40000000,1)C(277000000,4) C(317000000,5) =36.78%
C(40000000,4)C(277000000,1) C(317000000,5) =0.11% C(40000000,4)C(277000000,1) C(317000000,5) =0.11%
Review Exercises
2,4,7,11 2,4,7,11
13,103,1003,10003 13,103,1003,10003
The sequence is arithmetic. The common difference is d= 5 3 . d= 5 3 .
18,10,2,−6,−14 18,10,2,−6,−14
a 1 =−20, a n = a n−1 +10 a 1 =−20, a n = a n−1 +10
a n = 1 3 n+ 13 24 a n = 1 3 n+ 13 24
r=2 r=2
4, 16, 64, 256, 1024
3,12,48,192,768 3,12,48,192,768
a n =− 1 5 ⋅ ( 1 3 ) n−1 a n =− 1 5 ⋅ ( 1 3 ) n−1
∑ m=0 5 ( 1 2 m+5 ). ∑ m=0 5 ( 1 2 m+5 ).
S 11 =110 S 11 =110
S 9 ≈23.95 S 9 ≈23.95
S= 135 4 S= 135 4
$5,617.61
6
10 4 =10,000 10 4 =10,000
P(18,4)=73,440 P(18,4)=73,440
C( 15,6 )=5005 C( 15,6 )=5005
2 50 =1.13× 10 15 2 50 =1.13× 10 15
8! 3!2! =3360 8! 3!2! =3360
490,314 490,314
131,072 a 17 +1,114,112 a 16 b+4,456,448 a 15 b 2 131,072 a 17 +1,114,112 a 16 b+4,456,448 a 15 b 2
row: 1 | 2 | 3 | 4 | 5 | 6
row: 1 | 1,1 | 1,2 | 1,3 | 1,4 | 1,5 | 1,6
row: 2 | 2,1 | 2,2 | 2,3 | 2,4 | 2,5 | 2,6
row: 3 | 3,1 | 3,2 | 3,3 | 3,4 | 3,5 | 3,6
row: 4 | 4,1 | 4,2 | 4,3 | 4,4 | 4,5 | 4,6
row: 5 | 5,1 | 5,2 | 5,3 | 5,4 | 5,5 | 5,6
row: 6 | 6,1 | 6,2 | 6,3 | 6,4 | 6,5 | 6,6
1 6 1 6
5 9 5 9
4 9 4 9
1− C( 350,8 ) C( 500,8 ) ≈94.4% 1− C( 350,8 ) C( 500,8 ) ≈94.4%
C( 150,3 )C( 350,5 ) C( 500,8 ) ≈25.6% C( 150,3 )C( 350,5 ) C( 500,8 ) ≈25.6%
Practice Test
−14,−6,−2,0 −14,−6,−2,0
The sequence is arithmetic. The common difference is d=0.9. d=0.9.
a 1 =−2, a n = a n−1 − 3 2 ; a 22 =− 67 2 a 1 =−2, a n = a n−1 − 3 2 ; a 22 =− 67 2
The sequence is geometric. The common ratio is r= 1 2 . r= 1 2 .
a 1 =1, a n =− 1 2 ⋅ a n −1 a 1 =1, a n =− 1 2 ⋅ a n −1
∑ k=−3 15 ( 3 k 2 − 5 6 k ) ∑ k=−3 15 ( 3 k 2 − 5 6 k )
S 7 =−2604.2 S 7 =−2604.2
Total in account: $634,261.20; $634,261.20; Interest earned: $508,261.20 $508,261.20
5×3×2×3×2=180 5×3×2×3×2=180
C( 15,3 )=455 C( 15,3 )=455
10! 2!3!2! =151,200 10! 2!3!2! =151,200
429 x 14 16 429 x 14 16
4 7 4 7
5 7 5 7
C( 14,3 )C( 26,4 ) C( 40,7 ) ≈29.2% C( 14,3 )C( 26,4 ) C( 40,7 ) ≈29.2%