Slide 9 - 1 Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc. All rights reserved. Chapter 9 Sequences; Induction; the Binomial Theorem
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Slide 9 - 1Copyright © 2009 Pearson Education, Inc.
Active Learning Lecture SlidesFor use with Classroom Response Systems
© 2009 Pearson Education, Inc.All rights reserved.
Chapter 9Sequences;
Induction; the Binomial Theorem
Slide 9 - 2Copyright © 2009 Pearson Education, Inc.
Evaluate
a.
b.
c.
d.
252
1
20
3!7!
5!.
42
841
20
Slide 9 - 3Copyright © 2009 Pearson Education, Inc.
Evaluate
a.
b.
c.
d.
252
1
20
3!7!
5!.
42
841
20
Slide 9 - 4Copyright © 2009 Pearson Education, Inc.
Write out the first five terms of
a.
b.
c.
d.
s1 4, s2 5
3, s3
6
5, s4 1, s5
8
9
s1 4, s2 5
3, s3
6
5, s4 1, s5
8
9
sn 1 n 1 n 3
2n 1
.
s1 4, s2 5
3, s3
6
5, s4 1, s5
8
9
s1 4, s2 5
3, s3
6
5, s4 1, s5
8
9
Slide 9 - 5Copyright © 2009 Pearson Education, Inc.
Write out the first five terms of
a.
b.
c.
d.
s1 4, s2 5
3, s3
6
5, s4 1, s5
8
9
s1 4, s2 5
3, s3
6
5, s4 1, s5
8
9
sn 1 n 1 n 3
2n 1
.
s1 4, s2 5
3, s3
6
5, s4 1, s5
8
9
s1 4, s2 5
3, s3
6
5, s4 1, s5
8
9
Slide 9 - 6Copyright © 2009 Pearson Education, Inc.
Write down the nth term of the sequence {an} suggested by the pattern.
a.
c. an 1
n n 2 an 1
2n
an 1
n2nan n n 2 b.
d.
1
13,
1
24,
1
35,
1
4 6, ...
Slide 9 - 7Copyright © 2009 Pearson Education, Inc.
Write down the nth term of the sequence {an} suggested by the pattern.
a.
c. an 1
n n 2 an 1
2n
an 1
n2nan n n 2 b.
d.
1
13,
1
24,
1
35,
1
4 6, ...
Slide 9 - 8Copyright © 2009 Pearson Education, Inc.
Write out the first four terms of the sequence defined recursively by
a.
b.
c.
d.
a1 2, a2 4, a3 8, a4 16
a1 2, a2 6, a3 10, a4 18
a1 2; an 2an 1 2.
a1 2, a2 6, a3 14, a4 30
a1 2, a2 2, a3 2, a4 2
Slide 9 - 9Copyright © 2009 Pearson Education, Inc.
Write out the first four terms of the sequence defined recursively by
a.
b.
c.
d.
a1 2, a2 4, a3 8, a4 16
a1 2, a2 6, a3 10, a4 18
a1 2; an 2an 1 2.
a1 2, a2 6, a3 14, a4 30
a1 2, a2 2, a3 2, a4 2
Slide 9 - 10Copyright © 2009 Pearson Education, Inc.
Write out the sum
a.
b.
c.
d.
7 1115 ... 4n 1
3 7 11 ... 4n 1
4k 3 k0
n 1
.
3 7 11 ... 4n 3
7 1115 ... 4n 3
Slide 9 - 11Copyright © 2009 Pearson Education, Inc.
Write out the sum
a.
b.
c.
d.
7 1115 ... 4n 1
3 7 11 ... 4n 1
4k 3 k0
n 1
.
3 7 11 ... 4n 3
7 1115 ... 4n 3
Slide 9 - 12Copyright © 2009 Pearson Education, Inc.
Express the sum using summation notation
a.
c.
10k
ekk0
n
10k
ekk1
n
10
e
20
e2 30
e3 ...10n
en.
10 kekk1
n
10 kekk0
n
b.
d.
Slide 9 - 13Copyright © 2009 Pearson Education, Inc.
Express the sum using summation notation
a.
c.
10k
ekk0
n
10k
ekk1
n
10
e
20
e2 30
e3 ...10n
en.
10 kekk1
n
10 kekk0
n
b.
d.
Slide 9 - 14Copyright © 2009 Pearson Education, Inc.
Find the sum of the sequence
a. –40
b. –70
c. –61
d. –12
k k 10 k2
4
.
Slide 9 - 15Copyright © 2009 Pearson Education, Inc.
Find the sum of the sequence
a. –40
b. –70
c. –61
d. –12
k k 10 k2
4
.
Slide 9 - 16Copyright © 2009 Pearson Education, Inc.
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 b. $2080.03
c. $839.58 d. $683.95
Slide 9 - 17Copyright © 2009 Pearson Education, Inc.
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 b. $2080.03
c. $839.58 d. $683.95
Slide 9 - 18Copyright © 2009 Pearson Education, Inc.
Given the arithmetic sequenceFind the common difference and write out the first four terms.
a.
b.
c.
d.
d 1; s1 7, s2 6, s3 5, s4 4
d 1; s1 7, s2 6, s3 5, s4 4
sn n 8 ,
d 1; s1 7, s2 6, s3 5, s4 4
d 1; s1 7, s2 6, s3 5, s4 4
Slide 9 - 19Copyright © 2009 Pearson Education, Inc.
Given the arithmetic sequenceFind the common difference and write out the first four terms.
a.
b.
c.
d.
d 1; s1 7, s2 6, s3 5, s4 4
d 1; s1 7, s2 6, s3 5, s4 4
sn n 8 ,
d 1; s1 7, s2 6, s3 5, s4 4
d 1; s1 7, s2 6, s3 5, s4 4
Slide 9 - 20Copyright © 2009 Pearson Education, Inc.
Find the nth term and a21 of the arithmetic sequence {an} whose initial term a1 = 0 and
a.
b.
c.
d.
an 1
3n 1 ; a21
20
3
an 1
3n; a21 7
1
3.
an 1
3n 1 ; a21
22
3
an 3 n 1 ; a21 20
common difference d =
Slide 9 - 21Copyright © 2009 Pearson Education, Inc.
Find the nth term and a21 of the arithmetic sequence {an} whose initial term a1 = 0 and
a.
b.
c.
d.
an 1
3n 1 ; a21
20
3
an 1
3n; a21 7
1
3.
an 1
3n 1 ; a21
22
3
an 3 n 1 ; a21 20
common difference d =
Slide 9 - 22Copyright © 2009 Pearson Education, Inc.
Find the fifteenth term of the arithmetic sequence
a.
b.
c.
d.
73 3
79 3
11 3, 5 3, 1 3, ...
95 3
101 3
Slide 9 - 23Copyright © 2009 Pearson Education, Inc.
Find the fifteenth term of the arithmetic sequence
a.
b.
c.
d.
73 3
79 3
11 3, 5 3, 1 3, ...
95 3
101 3
Slide 9 - 24Copyright © 2009 Pearson Education, Inc.
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.
b.
c.
d.
a1 79, d 8, an an 1 8
a1 79, d 8, an an 1 8
a1 87, d 8, an an 1 8
a1 87, d 8, an an 1 8
Slide 9 - 25Copyright © 2009 Pearson Education, Inc.
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.
b.
c.
d.
a1 79, d 8, an an 1 8
a1 79, d 8, an an 1 8
a1 87, d 8, an an 1 8
a1 87, d 8, an an 1 8
Slide 9 - 26Copyright © 2009 Pearson Education, Inc.
Find the sum of 7 + 14 + 21 + … + 672.
a. 31,920
b. 32,256
c.
d. 32,592
65,863
2
Slide 9 - 27Copyright © 2009 Pearson Education, Inc.
Find the sum of 7 + 14 + 21 + … + 672.
a. 31,920
b. 32,256
c.
d. 32,592
65,863
2
Slide 9 - 28Copyright © 2009 Pearson Education, Inc.
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
Slide 9 - 29Copyright © 2009 Pearson Education, Inc.
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
Slide 9 - 30Copyright © 2009 Pearson Education, Inc.
Find the common ratio and write out the first
a.
b.
c.
d.
r 5
3; u1 5, u2
25
3, u3
125
9, u4
625
27
r 5
3; u1
5
3, u2
25
9, u3
125
27, u4
625
81
1
5.
3
n
n nu
r 5; u1 5, u2 25
3, u3
125
9, u4
625
27
r 5; u1 5, u2 25
3, u3
125
3, u4
625
3
four terms of the geometric sequence
Slide 9 - 31Copyright © 2009 Pearson Education, Inc.
Find the common ratio and write out the first
a.
b.
c.
d.
r 5
3; u1 5, u2
25
3, u3
125
9, u4
625
27
r 5
3; u1
5
3, u2
25
9, u3
125
27, u4
625
81
1
5.
3
n
n nu
r 5; u1 5, u2 25
3, u3
125
9, u4
625
27
r 5; u1 5, u2 25
3, u3
125
3, u4
625
3
four terms of the geometric sequence
Slide 9 - 32Copyright © 2009 Pearson Education, Inc.
Determine whether the sequence
a.
c.
Arithmetic; d 6
5Geometric; r
6
5
6
5
n
Geometric; r 5
6
b.
d. Neither
is arithmetic, geometric or neither. Find the common difference or common ratio.
Slide 9 - 33Copyright © 2009 Pearson Education, Inc.
Determine whether the sequence
a.
c.
Arithmetic; d 6
5Geometric; r
6
5
6
5
n
Geometric; r 5
6
b.
d. Neither
is arithmetic, geometric or neither. Find the common difference or common ratio.
Slide 9 - 34Copyright © 2009 Pearson Education, Inc.
Determine whether the sequence
a.
c.
Arithmetic; d 2 Geometric; r 5
5n2 2
Arithmetic; d 5
b.
d. Neither
is arithmetic, geometric or neither. Find the common difference or common ratio.
Slide 9 - 35Copyright © 2009 Pearson Education, Inc.
Determine whether the sequence
a.
c.
Arithmetic; d 2 Geometric; r 5
5n2 2
Arithmetic; d 5
b.
d. Neither
is arithmetic, geometric or neither. Find the common difference or common ratio.
Slide 9 - 36Copyright © 2009 Pearson Education, Inc.
Find the fifth term and the nth term of the geometric sequence whose initial term is a = 6 and common ratio is r = –5.
a.
b.
c.
d.
a5 750; an 6 5 n
a5 3750; an 6 5 n
a5 750; an 6 5 n 1
a5 3750; an 6 5 n 1
Slide 9 - 37Copyright © 2009 Pearson Education, Inc.
Find the fifth term and the nth term of the geometric sequence whose initial term is a = 6 and common ratio is r = –5.
a.
b.
c.
d.
a5 750; an 6 5 n
a5 3750; an 6 5 n
a5 750; an 6 5 n 1
a5 3750; an 6 5 n 1
Slide 9 - 38Copyright © 2009 Pearson Education, Inc.
Find the nth term of the geometric sequence5, –10, 20, –40, 80, … .
a.
b.
c.
d.
an 5 2 n 1
an 5 2 n
an a1 2n
an 5 2 n
Slide 9 - 39Copyright © 2009 Pearson Education, Inc.
Find the nth term of the geometric sequence5, –10, 20, –40, 80, … .
a.
b.
c.
d.
an 5 2 n 1
an 5 2 n
an a1 2n
an 5 2 n
Slide 9 - 40Copyright © 2009 Pearson Education, Inc.
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
Slide 9 - 41Copyright © 2009 Pearson Education, Inc.
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
Slide 9 - 42Copyright © 2009 Pearson Education, Inc.
Find the sum
a.
c.
700
81
2800
243
4
3
k1
k1
4
.
932
81
b.
d.12, 496
243
Slide 9 - 43Copyright © 2009 Pearson Education, Inc.
Find the sum
a.
c.
700
81
2800
243
4
3
k1
k1
4
.
932
81
b.
d.12, 496
243
Slide 9 - 44Copyright © 2009 Pearson Education, Inc.
Determine whether the infinite geometric series
a.
c.
Converges; 9 Converges; 3
6 2 2
3 ...
Converges; 26
3
b.
d. Diverges
converges, find its sum.
converges or diverges. If it
Slide 9 - 45Copyright © 2009 Pearson Education, Inc.
Determine whether the infinite geometric series
a.
c.
Converges; 9 Converges; 3
6 2 2
3 ...
Converges; 26
3
b.
d. Diverges
converges, find its sum.
converges or diverges. If it
Slide 9 - 46Copyright © 2009 Pearson Education, Inc.
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
Slide 9 - 47Copyright © 2009 Pearson Education, Inc.
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
Slide 9 - 48Copyright © 2009 Pearson Education, Inc.
Evaluate the expression
a. 1848
b. 924
c. 665,280
d. 462
12
6
.
Slide 9 - 49Copyright © 2009 Pearson Education, Inc.
Evaluate the expression
a. 1848
b. 924
c. 665,280
d. 462
12
6
.
Slide 9 - 50Copyright © 2009 Pearson Education, Inc.
Expand
a.
b.
c.
d.
32x5 810x4 1080x3 1080x2 810x 243
32x5 48x4 72x3 108x2 162x 243
2x 3 5
4x2 12x 9 5
32x5 240x4 720x3 1080x2 810x 243
using the binomial theorem.
Slide 9 - 51Copyright © 2009 Pearson Education, Inc.
Expand
a.
b.
c.
d.
32x5 810x4 1080x3 1080x2 810x 243
32x5 48x4 72x3 108x2 162x 243
2x 3 5
4x2 12x 9 5
32x5 240x4 720x3 1080x2 810x 243
using the binomial theorem.
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