Transcript ch.9 active learning
Active Learning Lecture Slides
For use with Classroom Response Systems
Chapter 9 Sequences; Induction; the Binomial Theorem
© 2009 Pearson Education, Inc.
All rights reserved.
Copyright © 2009 Pearson Education, Inc.
Slide 9 - 1
Evaluate 3!7!
.
5!
a.
b.
c.
d.
252 1 20 42 841 20 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 2
Evaluate 3!7!
.
5!
a.
b.
c.
d.
252 1 20 42 841 20 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 3
Write out the first five terms of 1
n
1
n
2
n
3 1 .
a.
b.
c.
d.
s s s
1 1 1 4, 4,
s
1 4, 4,
s
2
s
2
s
2
s
2 5 3 , 5 3 , 5 3 , 5 3 ,
s
3
s
3 6 5 , 6 5 ,
s s
3 3 6 5 , 6 5 ,
s
4
s s
4 4
s
4 1, 1, 1, 1,
s
5
s
5
s
5
s
5 8 9 8 9 8 9 8 9
Slide 9 - 4
Copyright © 2009 Pearson Education, Inc.
Write out the first five terms of 1
n
1
n
2
n
3 1 .
a.
b.
c.
d.
s s s
1 1 1 4, 4,
s
1 4, 4,
s
2
s
2
s
2
s
2 5 3 , 5 3 , 5 3 , 5 3 ,
s
3
s
3 6 5 , 6 5 ,
s s
3 3 6 5 , 6 5 ,
s
4
s s
4 4
s
4 1, 1, 1, 1,
s
5
s
5
s
5
s
5 8 9 8 9 8 9 8 9
Slide 9 - 5
Copyright © 2009 Pearson Education, Inc.
a.
Write down the nth term of the sequence {
a n
} suggested by the pattern.
1 1 3 , 1 2 4 , 1 3 5 , 4 1 6 , ...
a n
2 b.
a n
1
n
2
n
c.
a n
1 2
n
Copyright © 2009 Pearson Education, Inc.
d.
a n
1 2
Slide 9 - 6
a.
Write down the nth term of the sequence {
a n
} suggested by the pattern.
1 1 3 , 1 2 4 , 1 3 5 , 4 1 6 , ...
a n
2 b.
a n
1
n
2
n
c.
a n
1 2
n
Copyright © 2009 Pearson Education, Inc.
d.
a n
1 2
Slide 9 - 7
Write out the first four terms of the sequence defined recursively by
a
1 2;
a n
2
a n
1 2.
a.
b.
c.
d.
a
1 2,
a
2 4,
a
3 8,
a
4 16
a
1 2,
a
2 6,
a
3 10,
a
4 18
a
1 2,
a
2 6,
a
3 14,
a
4 30
a
1 2,
a
2 2,
a
3 2,
a
4 2 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 8
Write out the first four terms of the sequence defined recursively by
a
1 2;
a n
2
a n
1 2.
a.
b.
c.
d.
a
1 2,
a
2 4,
a
3 8,
a
4 16
a
1 2,
a
2 6,
a
3 10,
a
4 18
a
1 2,
a
2 6,
a
3 14,
a
4 30
a
1 2,
a
2 2,
a
3 2,
a
4 2 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 9
Write out the sum
n
1
k
0 4
k
3 .
a.
b.
c.
d.
7 11 15 ...
4
n
1 3 7 11 ...
4
n
1 3 7 11 ...
4
n
3 7 11 15 ...
4
n
3 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 10
Write out the sum
n
1
k
0 4
k
3 .
a.
b.
c.
d.
7 11 15 ...
4
n
1 3 7 11 ...
4
n
1 3 7 11 ...
4
n
3 7 11 15 ...
4
n
3 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 11
Express the sum using summation notation 10
e
20
e
2 30
e
3 ...
10
n e n
.
a.
n
k
0 10
k e k
b.
n
k
1 10
k e k
c.
n
k
1 10
e k
k
Copyright © 2009 Pearson Education, Inc.
d.
n
k
0 10
e k
k
Slide 9 - 12
Express the sum using summation notation 10
e
20
e
2 30
e
3 ...
10
n e n
.
a.
n
k
0 10
k e k
b.
n
k
1 10
k e k
c.
n
k
1 10
e k
k
Copyright © 2009 Pearson Education, Inc.
d.
n
k
0 10
e k
k
Slide 9 - 13
Find the sum of the sequence 4
k
2 10 .
a.
–40 b.
–70 c.
–61 d.
–12 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 14
Find the sum of the sequence 4
k
2 10 .
a.
–40 b.
–70 c.
–61 d.
–12 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 15
After working for 25 years you would like to have $600,000 in an annuity for early retirement. If the annuity interest rate is 7.5%, compounded monthly, what will your monthly deposit need to be?
a.
$1373.90
c.
$839.58
Copyright © 2009 Pearson Education, Inc.
b.
$2080.03
d.
$683.95
Slide 9 - 16
After working for 25 years you would like to have $600,000 in an annuity for early retirement. If the annuity interest rate is 7.5%, compounded monthly, what will your monthly deposit need to be?
a.
$1373.90
c.
$839.58
Copyright © 2009 Pearson Education, Inc.
b.
$2080.03
d.
$683.95
Slide 9 - 17
Given the arithmetic sequence
n
8 , Find the common difference and write out the first four terms.
a.
d
1;
s
1 7,
s
2 6,
s
3 5,
s
4 4 b.
d
1;
s
1 7,
s
2 6,
s
3 5,
s
4 4 c.
d
1;
s
1 7,
s
2 6,
s
3 5,
s
4 4 d.
d
1;
s
1 7, Copyright © 2009 Pearson Education, Inc.
s
2 6,
s
3 5,
s
4 4
Slide 9 - 18
Given the arithmetic sequence
n
8 , Find the common difference and write out the first four terms.
a.
d
1;
s
1 7,
s
2 6,
s
3 5,
s
4 4 b.
d
1;
s
1 7,
s
2 6,
s
3 5,
s
4 4 c.
d
1;
s
1 7,
s
2 6,
s
3 5,
s
4 4 d.
d
1;
s
1 7, Copyright © 2009 Pearson Education, Inc.
s
2 6,
s
3 5,
s
4 4
Slide 9 - 19
Find the
n
th term and
a
21 of the arithmetic sequence {
a n
} whose initial term common difference
d
= 1 3 .
a
1 = 0 and a.
b.
c.
a n a n a n
1 1 3
n n
; 3 1 3
n
1 ;
a
21 1 ;
a
21 7
a
21 20 3 22 3 d.
a n
3
n
1 ;
a
21 20
Slide 9 - 20
Copyright © 2009 Pearson Education, Inc.
Find the
n
th term and
a
21 of the arithmetic sequence {
a n
} whose initial term common difference
d
= 1 3 .
a
1 = 0 and a.
b.
c.
a n a n a n
1 1 3
n n
; 3 1 3
n
1 ;
a
21 1 ;
a
21 7
a
21 20 3 22 3 d.
a n
3
n
1 ;
a
21 20
Slide 9 - 21
Copyright © 2009 Pearson Education, Inc.
Find the fifteenth term of the arithmetic sequence 11 3, 5 3, 1 3, ...
a.
73 3 b.
79 3 c.
95 3 d.
101 3 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 22
Find the fifteenth term of the arithmetic sequence 11 3, 5 3, 1 3, ...
a.
73 3 b.
79 3 c.
95 3 d.
101 3 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 23
Find the first term, the common difference, and give a recursive formula for the arithmetic sequence whose 7th term is 31 and 16th term is –41.
a.
a
1 79,
d
8,
a n
a n
1 8 b.
a
1 79,
d
8,
a n
a n
1 8 c.
a
1 87,
d
8,
a n
a n
1 8 d.
a
1 87, Copyright © 2009 Pearson Education, Inc.
d
8,
a n
a n
1 8
Slide 9 - 24
Find the first term, the common difference, and give a recursive formula for the arithmetic sequence whose 7th term is 31 and 16th term is –41.
a.
a
1 79,
d
8,
a n
a n
1 8 b.
a
1 79,
d
8,
a n
a n
1 8 c.
a
1 87,
d
8,
a n
a n
1 8 d.
a
1 87, Copyright © 2009 Pearson Education, Inc.
d
8,
a n
a n
1 8
Slide 9 - 25
Find the sum of 7 + 14 + 21 + … + 672.
a.
31,920 b.
32,256 c.
65, 863 2 d.
32,592 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 26
Find the sum of 7 + 14 + 21 + … + 672.
a.
31,920 b.
32,256 c.
65, 863 2 d.
32,592 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 27
A theater has 20 rows with 24 seats in the first row, 28 in the second row, 32 in the third row, and so forth. How many seats are in the theater?
a.
2480 seats b.
1280 seats c.
1240 seats d.
2560 seats Copyright © 2009 Pearson Education, Inc.
Slide 9 - 28
A theater has 20 rows with 24 seats in the first row, 28 in the second row, 32 in the third row, and so forth. How many seats are in the theater?
a.
2480 seats b.
1280 seats c.
1240 seats d.
2560 seats Copyright © 2009 Pearson Education, Inc.
Slide 9 - 29
Find the common ratio and write out the first 5
n
four terms of the geometric sequence
u n
3
n
1 .
a.
b.
r
r
5 3 ; 5 3 ;
u
1
u
1 5, 5 3 ,
u
2
u
2 25 , 3 25 , 9
u
3 125 , 9
u
3 125 , 27
u
4
u
4 625 27 625 81 c.
d.
r
5;
r
5;
u
1 5,
u
1 5, Copyright © 2009 Pearson Education, Inc.
u
2
u
2 25 , 3 25 , 3
u
3 125 , 9
u
3 125 , 3
u
4
u
4 625 27 625 3
Slide 9 - 30
Find the common ratio and write out the first 5
n
four terms of the geometric sequence
u n
3
n
1 .
a.
b.
r
r
5 3 ; 5 3 ;
u
1
u
1 5, 5 3 ,
u
2
u
2 25 , 3 25 , 9
u
3 125 , 9
u
3 125 , 27
u
4
u
4 625 27 625 81 c.
d.
r
5;
r
5;
u
1 5,
u
1 5, Copyright © 2009 Pearson Education, Inc.
u
2
u
2 25 , 3 25 , 3
u
3 125 , 9
u
3 125 , 3
u
4
u
4 625 27 625 3
Slide 9 - 31
Determine whether the sequence 6 5
n
is arithmetic, geometric or neither. Find the common difference or common ratio.
a.
Arithmetic;
d
6 5 b.
Geometric;
r
6 5 c.
Geometric;
r
5 6 Copyright © 2009 Pearson Education, Inc.
d.
Neither
Slide 9 - 32
Determine whether the sequence 6 5
n
is arithmetic, geometric or neither. Find the common difference or common ratio.
a.
Arithmetic;
d
6 5 b.
Geometric;
r
6 5 c.
Geometric;
r
5 6 Copyright © 2009 Pearson Education, Inc.
d.
Neither
Slide 9 - 33
Determine whether the sequence 5
n
2 2 is arithmetic, geometric or neither. Find the common difference or common ratio.
a.
Arithmetic;
d
2 b.
Geometric;
r
5 c.
Arithmetic;
d
5 d.
Neither Copyright © 2009 Pearson Education, Inc.
Slide 9 - 34
Determine whether the sequence 5
n
2 2 is arithmetic, geometric or neither. Find the common difference or common ratio.
a.
Arithmetic;
d
2 b.
Geometric;
r
5 c.
Arithmetic;
d
5 d.
Neither Copyright © 2009 Pearson Education, Inc.
Slide 9 - 35
Find the fifth term and the
n
th term of the geometric sequence whose initial term is
a
= 6 and common ratio is
r
= –5.
a.
a
5 750;
a n
6
n
b.
a
5 3750;
a n
6
n
c.
a
5 750;
a n
6
n
1 d.
a
5 3750; Copyright © 2009 Pearson Education, Inc.
a n
6
n
1
Slide 9 - 36
Find the fifth term and the
n
th term of the geometric sequence whose initial term is
a
= 6 and common ratio is
r
= –5.
a.
a
5 750;
a n
6
n
b.
a
5 3750;
a n
6
n
c.
a
5 750;
a n
6
n
1 d.
a
5 3750; Copyright © 2009 Pearson Education, Inc.
a n
6
n
1
Slide 9 - 37
Find the
n
th term of the geometric sequence 5, –10, 20, –40, 80, … .
a.
b.
a n
5
n
1
a n
5
n
c.
a n
a
1 2
n
d.
a n
5
n
Copyright © 2009 Pearson Education, Inc.
Slide 9 - 38
Find the
n
th term of the geometric sequence 5, –10, 20, –40, 80, … .
a.
b.
a n
5
n
1
a n
5
n
c.
a n
a
1 2
n
d.
a n
5
n
Copyright © 2009 Pearson Education, Inc.
Slide 9 - 39
A new piece of equipment cost a company $48,000. Each year, for tax purposes, the company depreciates the value by 25%. What value should the company give the equipment after 7 years?
a.
$3 b.
$6407 c.
$8543 d.
$12 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 40
A new piece of equipment cost a company $48,000. Each year, for tax purposes, the company depreciates the value by 25%. What value should the company give the equipment after 7 years?
a.
$3 b.
$6407 c.
$8543 d.
$12 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 41
Find the sum 4
k
1 4 3
k
1 .
a.
700 81 c.
932 81 Copyright © 2009 Pearson Education, Inc.
b.
2800 243 d.
12, 496 243
Slide 9 - 42
Find the sum 4
k
1 4 3
k
1 .
a.
700 81 c.
932 81 Copyright © 2009 Pearson Education, Inc.
b.
2800 243 d.
12, 496 243
Slide 9 - 43
Determine whether the infinite geometric series 6 2 2 3 ...
converges or diverges. If it converges, find its sum.
a.
b.
Converges; 3 c.
Converges; 26 3 Copyright © 2009 Pearson Education, Inc.
d.
Diverges
Slide 9 - 44
Determine whether the infinite geometric series 6 2 2 3 ...
converges or diverges. If it converges, find its sum.
a.
b.
Converges; 3 c.
Converges; 26 3 Copyright © 2009 Pearson Education, Inc.
d.
Diverges
Slide 9 - 45
A pendulum bob swings through an arc 80 inches long on its first swing. Each swing thereafter, it swings only 90% as far as on the previous swing. How far will it swing altogether before coming to a complete stop?
a.
89 inches b.
800 inches c.
400 inches d.
178 inches Copyright © 2009 Pearson Education, Inc.
Slide 9 - 46
A pendulum bob swings through an arc 80 inches long on its first swing. Each swing thereafter, it swings only 90% as far as on the previous swing. How far will it swing altogether before coming to a complete stop?
a.
89 inches b.
800 inches c.
400 inches d.
178 inches Copyright © 2009 Pearson Education, Inc.
Slide 9 - 47
Evaluate the expression 12 6 .
a.
1848 b.
924 c.
665,280 d.
462 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 48
Evaluate the expression 12 6 .
a.
1848 b.
924 c.
665,280 d.
462 Copyright © 2009 Pearson Education, Inc.
Slide 9 - 49
Expand 2
x
3 5 using the binomial theorem.
a.
32
x
5 810
x
4 1080
x
3 1080
x
2 810
x
243 b.
32
x
5 48
x
4 72
x
3 108
x
2 162
x
243 c.
4
x
2 12
x
9 5 d.
32
x
5 240
x
4 720
x
3 1080
x
2 810
x
243
Slide 9 - 50
Copyright © 2009 Pearson Education, Inc.
Expand 2
x
3 5 using the binomial theorem.
a.
32
x
5 810
x
4 1080
x
3 1080
x
2 810
x
243 b.
32
x
5 48
x
4 72
x
3 108
x
2 162
x
243 c.
4
x
2 12
x
9 5 d.
32
x
5 240
x
4 720
x
3 1080
x
2 810
x
243
Slide 9 - 51
Copyright © 2009 Pearson Education, Inc.