Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
3 2
3 1
n
n
3 3 1 3 2n n n
1/26/2015 Precalculus HWQ:
Simplify the factorial expression:
!!
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3
Precalculus Warm-up
Write an expression for the apparent nth term of the sequence:
1 3 7 15 311 ,1 ,1 ,1 ,1 ,...
2 4 8 16 32
1
2 11
2
2 1
2
n
n n
n
n
a
or
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
An infinite sequence is a function whose domain is the set of positive integers.
a1, a2, a3, a4, . . . , an, . . .
The first three terms of the sequence an = 4n – 7 are
a1 = 4(1) – 7 = – 3
a2 = 4(2) – 7 = 1
a3 = 4(3) – 7 = 5.
finite sequence
terms
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
A sequence is arithmetic if the differences between consecutive terms are the same.
4, 9, 14, 19, 24, . . .
9 – 4 = 5
14 – 9 = 5
19 – 14 = 5
24 – 19 = 5
arithmetic sequence
The common difference, d, is 5.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
Example: Find the first five terms of the sequence and determine if it is arithmetic.
an = 1 + (n – 1)4
This is an arithmetic sequence.
d = 4
a1 = 1 + (1 – 1)4 = 1 + 0 = 1
a2 = 1 + (2 – 1)4 = 1 + 4 = 5
a3 = 1 + (3 – 1)4 = 1 + 8 = 9
a4 = 1 + (4 – 1)4 = 1 + 12 = 13
a5 = 1 + (5 – 1)4 = 1 + 16 = 17
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7
Determine whether or not each sequence is arithmetic.
a) -12, -7, -2, 3, 8, . . .
b) ln1, ln2, ln3, ln4, ln5, . . .
1 2 4 8 16c) , , , , ,...
3 3 3 3 3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8
The nth term of an arithmetic sequence has the form an = a1 + (n – 1)d
or the alternate form: an = dn + c
where d is the common difference and c = a1 – d.
2, 8, 14, 20, 26, . . . .
d = 8 – 2 = 6
a1 = 2
c = 2 – 6 = – 4
The nth term is 6n – 4.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
a1 – d =
Example: Find the formula for the nth term of an arithmetic sequence whose common difference is 4 and whose first term is 15. Find the first five terms of the sequence.
an = dn + c
= 4n + 11
15,
d = 4
a1 = 15 19, 23, 27, 31.
The first five terms are
15 – 4 = 11
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
Graphing Utility: Find the first 5 terms of the arithmetic sequence an = 4n + 11.
List Menu:
variable beginning value
end value
• Example: Find the formula for the nth term of an arithmetic sequence whose 4th term is 18 and whose 13th term is 63. Find the 20th term of the sequence.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
18 9 63d
9 45d
5d
5 2na n
20 98a
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12
You Try: Find the formula for the nth term of an arithmetic sequence whose 10th term is 32 and whose 16th term is 50. Find the 30th term of the sequence.
3 2na n 30 92a
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13
Try another: Find the formula for the nth term of an arithmetic sequence whose 5th term is 190 and whose 10th term is 115. Find the 15th term of the sequence.
15 265na n 15 40a
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14
Try another: Find the formula for the nth term of an arithmetic sequence whose 10th term is -330 and whose 20th term is -450. Find the 52nd term of the sequence.
Homework Day 1
• Pg. 573 1-41 odds only
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18
1/27/2015 Precalculus Warm-up :
Find a formula for the arithmetic sequence where 5 1519 89a and a
7 16na n
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19
Precalculus Warm-up:
Simplify the factorial expression:
2 3 !
2 2 !
n
n
1
2 2 2 1 2 2 1 2 2n n n n n
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20
The sum of the first n terms of a sequence is represented by summation notation.
1 2 3 41
n
i ni
a a a a a a
index of summation
upper limit of summation
lower limit of summation
5
1
1i
n
(11) (1 2) (1 3) (1 4) (15)
2 3 4 5 6
20
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21
Consider the infinite sequence a1, a2, a3, . . ., ai, . . ..
1. The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence.
1
n
ii
a
a1 + a2 + a3 + . . . + an
2. The sum of all the terms of the infinite sequence is called an infinite series.
1i
i
a
a1 + a2 + a3 + . . . + ai + . . .
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22
Example:
Find the sum: 10
1
5n
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23
The sum of a finite arithmetic sequence with n terms is given by
1( ).2n nnS a a
5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = ?
( )501 2755 )0 5(552nS
n = 10
a1 = 5 a10 = 50
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24
The sum of the first n terms of an infinite sequence is called the nth partial sum.
1( )2n nnS a a
( )190 25(184) 4602
50 6 0nS
an = dn + c = 4n – 10
Example: Find the 50th partial sum of the arithmetic sequence – 6, – 2, 2, 6, . . .
a50 = 4(50) – 10 = 190
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25
100
1
2i
n
Example: Find the partial sum.
2( ) 2( ) 2( ) 2( )1 2 3 100 2 4 6 200
a1 a100
100 ( )2
20 01 00 2S
50(202) 10,100
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26
In an arithmetic sequence, the 20th term is 116
and the 24th term is 140.
Find the sum of the first 50 terms.
6 4na n
50 7450S
1 502, 296a a
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27
In an arithmetic sequence, the 12th term is 25
and the 30th term is 97.
Find the sum of the first 40 terms.
4 23na n
40 2360S
1 4019, 137a a
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28
Graphing Utility: Find the first 5 terms of the arithmetic sequence an = 4n + 11.
List Menu:
variable beginning value
end value
Graphing Utility: Find the sum 100
1
2 .i
n
List Menu:
lower limit
upper limit
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29
• Example: Find the 150th partial sum of the sequence: 5, 16, 27, 38, 49, …
150 123,675S
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 30
A stadium has 20 rows of seats. There are
20 seats in row 1, 21 in row 2, 22 in row 3,
etc. How many total seats are there?
590
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 31
Find a formula to represent the sum of n positive odd integers.
21 2 1 22 2n
n nS n n n
Homework Day 2
• Pg. 573 43-81 odds only
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 32
Top Related