Transcript 8.2 Arithmetic Sequences and Series

Digital Lesson
Arithmetic Sequences
and Series
2015
2 days
1/26/2015 Precalculus HWQ:
Simplify the factorial expression:
 3n  2 !
 3n  1!
3n 3n 13n  2
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2
Precalculus Warm-up
Write an expression for the apparent
nth term of the sequence:
1
3
7
15
31
1  ,1  ,1  ,1  ,1  ,...
2
4
8
16
32
2n  1
an  1  n
2
2 n 1  1
or
2n
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3
An infinite sequence is a function whose domain
is the set of positive integers.
a1, a2, a3, a4, . . . , an, . . .
terms
The first three terms of the sequence an = 4n – 7 are
a1 = 4(1) – 7 = – 3
a2 = 4(2) – 7 = 1
finite sequence
a3 = 4(3) – 7 = 5.
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4
A sequence is arithmetic if the differences
between consecutive terms are the same.
4, 9, 14, 19, 24, . . .
arithmetic sequence
9–4=5
14 – 9 = 5
19 – 14 = 5
The common difference, d, is 5.
24 – 19 = 5
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5
Example: Find the first five terms of the sequence
and determine if it is arithmetic.
an = 1 + (n – 1)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
d=4
a4 = 1 + (4 – 1)4 = 1 + 12 = 13
a5 = 1 + (5 – 1)4 = 1 + 16 = 17
This is an arithmetic sequence.
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6
Determine whether or not each sequence is arithmetic.
a) -12, -7, -2, 3, 8, . . .
b) ln1, ln2, ln3, ln4, ln5, . . .
1 2 4 8 16
c) , , , , ,...
3 3 3 3 3
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7
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.
a1 = 2
2, 8, 14, 20, 26, . . . .
c=2–6=–4
d=8–2=6
The nth term is 6n – 4.
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8
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
a1 – d = 15 – 4 = 11
= 4n + 11
The first five terms are
a1 = 15
15, 19, 23, 27, 31.
d=4
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9
Graphing Utility: Find the first 5 terms of the arithmetic
sequence an = 4n + 11.
beginning
variable
value
List Menu:
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end value
10
• 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.
18  9d  63
an  5n  2
9d  45
a20  98
d 5
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11
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.
an  3n  2
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a30  92
12
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.
an  15n  265
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a15  40
13
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.
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14
56
Ex: Find the sum:
i
i 1
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15
Homework Day 1
• Pg. 573
1-41 odds only
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16
Digital Lesson
Arithmetic Sequences
and Series Day 2
2015
1/27/2015 Precalculus Warm-up :
Find a formula for the arithmetic sequence
where a5 19 and a15  89
an  7n 16
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18
Precalculus Warm-up:
Simplify the factorial expression:
 2n  3!
 2n  2!
1
 2n  2 2n 1 2n  2n  1 2n  2 
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19
The sum of the first n terms of a sequence is
represented by summation notation.
upper limit of summation
n
a  a  a
i 1
i
1
2
 a3  a4 
 an
lower limit of summation
index of
summation
5
 1  n   (1 1)  (1 2)  (1 3)  (1 4)  (1 5)
i 1
 2  3 4 5 6
 20
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20
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.
n
a1 + a2 + a3 + . . . + an   ai
i 1
2. The sum of all the terms of the infinite sequence is
called an infinite series.

a1 + a2 + a3 + . . . + ai + . . .   ai
i 1
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21
Example:
Find the sum:
10
 5n
1
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22
The sum of a finite arithmetic sequence with n
terms is given by
Sn  n (a1  an).
2
5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = ?
n = 10
a1 = 5
a10 = 50
Sn  10 (5  50)  5(55)  275
2
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23
The sum of the first n terms of an infinite sequence
is called the nth partial sum.
Sn  n (a1  an)
2
Example: Find the 50th partial sum of the arithmetic
sequence – 6, – 2, 2, 6, . . .
an = dn + c = 4n – 10
a50 = 4(50) – 10 = 190
Sn  50 (6  190)  25(184)  4600
2
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Example: Find the partial sum.
100
  2n   2(1)  2(2)  2(3) 
 246
i 1
 2(100)
 200
a1
a100
S100  100 (2  200)
2
 50(202)  10,100
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25
In an arithmetic sequence, the 20th term is 116
and the 24th term is 140.
Find the sum of the first 50 terms.
an  6n  4
a1  2, a50  296
S50  7450
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26
In an arithmetic sequence, the 12th term is 25
and the 30th term is 97.
Find the sum of the first 40 terms.
an  4n  23
a1  19, a40  137
S40  2360
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Graphing Utility: Find the first 5 terms of the arithmetic
sequence an = 4n + 11.
beginning
variable
value
end value
List Menu:
100
Graphing Utility: Find the sum   2n .
i 1
List Menu:
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lower limit
upper
limit
28
• Example: Find the 150th partial sum of the
sequence: 5, 16, 27, 38, 49, …
S150  123, 675
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29
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
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30
Find a formula to represent the sum of n
positive odd integers.
n
n
Sn  1   2n  1    2n   n2
2
2
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31
Homework Day 2
• Pg. 573
43-81 odds only
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32
1/27/2015 Precalculus HWQ:

Find the fourth partial sum of
4
 
i
1 .
5

2
i 1
    
 5 1   5 1   5 1   5 1 
2
4
8
16
i
1
2
3
1  5 1 5 1 5 1 5 1
5

2
2
2
2
2
i1
4
 555 5
2 4 8 16
 40  20  10  5  75
16 16 16 16 16
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