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 a n 293⋅1.026 a n
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
9.1 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
9.2 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 =1+2n
a n =−105+100n a n =−105+100n
a n =1.8n a n =1.8n
a n =13.1+2.7n a n =13.1+2.7n
a n = 1 3 n− 1 3 a n = 1 3 n− 1 3
There are 10 terms in the sequence.
There are 6 terms in the sequence.
The graph does not represent an arithmetic sequence.
1,4,7,10,13,16,19 1,4,7,10,13,16,19
Answers will vary. Examples: a n =20.6n a n =20.6n and a n =2+20.4n. a n =2+20.4n.
a 11 =−17a+38b a 11 =−17a+38b
The sequence begins to have negative values at the 13th term, a 13 =− 1 3 a 13 =− 1 3
Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: a 1 =3, a n = a n−1 −3. a 1 =3, a n = a n−1 −3. First 4 terms: 3,0,−3,−6 a 31 =−87 3,0,−3,−6 a 31 =−87
9.3 Section Exercises
A sequence in which the ratio between any two consecutive terms is constant.
Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.
Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.
The common ratio is −2 −2
The sequence is geometric. The common ratio is 2.
The sequence is geometric. The common ratio is − 1 2 . − 1 2 .
The sequence is geometric. The common ratio is 5. 5.
5,1, 1 5 , 1 25 , 1 125 5,1, 1 5 , 1 25 , 1 125
800,400,200,100,50 800,400,200,100,50
a 4 =− 16 27 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
9.4 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
ਲੜੀ ਪਰਿਭਾਸ਼ਿਤ ਹੈ। S= 2 1−0.8 S= 2 1−0.8
ਲੜੀ ਪਰਿਭਾਸ਼ਿਤ ਹੈ। S= −1 1−( − 1 2 ) S= −1 1−( − 1 2 )
S n S n ਦਾ ਗ੍ਰਾਫ 1 ਦੇ ਨੇੜੇ ਪਹੁੰਚਦਾ ਪ੍ਰਤੀਤ ਹੁੰਦਾ ਹੈ। ਇਹ ਸਮਝ ਆਉਂਦਾ ਹੈ ਕਿਉਂਕਿ ∑ k=1 ∞ ( 1 2 ) k ∑ k=1 ∞ ( 1 2 ) k ਇੱਕ ਪਰਿਭਾਸ਼ਿਤ ਅਨੰਤ ਜਿਓਮੈਟ੍ਰਿਕ ਲੜੀ ਹੈ ਜਿਸਦਾ S= 1 2 1–( 1 2 ) =1 ਹੈ। S= 1 2 1–( 1 2 ) =1 ਹੈ।
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 ਪਦ
r= 4 5 r= 4 5
$400 ਪ੍ਰਤੀ ਮਹੀਨਾ
420 feet
12 feet
9.5 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
9.6 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 ਏ 3 ਬੀ 16 1,082,565 ਏ 3 ਬੀ 16
1152 ਵਾਈ 2 ਐਕਸ 7 1152 ਵਾਈ 2 ਐਕਸ 7
f 2 (x)= x 4 +12 x 3
f 4 (x)= x 4 +12 x 3 +54 x 2 +108x
590,625 x 5 y 2
k−1
ਬਿਨੋਮੀਅਲ ਥਿਊਰਮ ਦੀ ਵਰਤੋਂ ਕਰਕੇ ( x 3 +2 y 2 −z) 5 ਸਮੀਕਰਨ ਦਾ ਵਿਸਤਾਰ ਨਹੀਂ ਕੀਤਾ ਜਾ ਸਕਦਾ ਕਿਉਂਕਿ ਇਸਨੂੰ ਦੋ-ਪਦੀ (binomial) ਵਜੋਂ ਮੁੜ-ਲਿਖਿਆ ਨਹੀਂ ਜਾ ਸਕਦਾ।
9.7 ਭਾਗ ਅਭਿਆਸ
ਸੰਭਾਵਨਾ; ਕਿਸੇ ਘਟਨਾ ਦੀ ਸੰਭਾਵਨਾ 0 ਅਤੇ 1 ਦੇ ਵਿਚਕਾਰ ਦੇ ਮੁੱਲਾਂ ਤੱਕ ਸੀਮਤ ਹੈ, ਜਿਸ ਵਿੱਚ 0 ਅਤੇ 1 ਸ਼ਾਮਲ ਹਨ।
ਪ੍ਰਯੋਗ ਇੱਕ ਗਤੀਵਿਧੀ ਹੈ ਜਿਸਦਾ ਨਤੀਜਾ ਦੇਖਿਆ ਜਾ ਸਕਦਾ ਹੈ।
ਦੋ ਘਟਨਾਵਾਂ ਦੇ ਸੰਯੋਗ (union) ਦੀ ਸੰਭਾਵਨਾ ਇੱਕ ਅਜਿਹੀ ਸੰਖਿਆ ਹੈ ਜੋ ਕਿਸੇ ਸੰਭਾਵਨਾ ਮਾਡਲ ਤੋਂ ਘੱਟੋ-ਘੱਟ ਇੱਕ ਘਟਨਾ ਦੇ ਵਾਪਰਨ ਦੀ ਸੰਭਾਵਨਾ ਦਾ ਵਰਣਨ ਕਰਦੀ ਹੈ। ਸੈੱਟਾਂ ਦੇ ਸੰਯੋਗ A ਅਤੇ B ਅਤੇ ਘਟਨਾਵਾਂ ਦੇ ਸੰਯੋਗ A ਅਤੇ B, ਦੋਵਾਂ ਵਿੱਚ, ਸੰਯੋਗ ਵਿੱਚ ਜਾਂ ਤਾਂ A ਜਾਂ B ਜਾਂ ਦੋਵੇਂ ਸ਼ਾਮਲ ਹੁੰਦੇ ਹਨ। ਫਰਕ ਇਹ ਹੈ ਕਿ ਸੈੱਟਾਂ ਦਾ ਸੰਯੋਗ ਇੱਕ ਹੋਰ ਸੈੱਟ ਦਾ ਨਤੀਜਾ ਦਿੰਦਾ ਹੈ, ਜਦੋਂ ਕਿ ਘਟਨਾਵਾਂ ਦਾ ਸੰਯੋਗ ਇੱਕ ਸੰਭਾਵਨਾ ਹੁੰਦੀ ਹੈ, ਇਸ ਲਈ ਇਹ ਹਮੇਸ਼ਾ 0 ਅਤੇ 1 ਦੇ ਵਿਚਕਾਰ ਇੱਕ ਸੰਖਿਆਤਮਕ ਮੁੱਲ ਹੁੰਦਾ ਹੈ।
1 2
5 8
1 2
3 8
1 4
3 4
3 8
1 8
15 16
5 8
1 13
1 26
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%