sequences and series on occasion, it is convenient to begin subscripting a sequence with 0 instead...
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Sequences and Series
On occasion, it is convenient to begin subscripting a sequence with 0 instead of 1 so that the terms of the sequence become
0 1 2 3, , , ,...a a a a
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Example 1.
Write the first four terms of the sequences given by
a. 3 2 b. 3 1n
n na n a
a. 1 3 1 2 1a
2 3 2 2 4a
3 3 3 2 7a 4 3 4 2 10a
b. 1
1 3 1 2a
2
2 3 1 4a 3
3 3 1 2a
4
4 3 1 4a
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Example 2.
1Write the first five terms of the sequence given by
2 1
n
na n
1
1
1 11
2 1 1 1a
2
2
1 1
2 2 1 3a
3
3
1 1
2 3 1 5a
4
4
1 1
2 4 1 7a
5
5
1 1
2 5 1 9a
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Simply listing the first few terms is not sufficient to define a unique sequence-----the nth term must be given.
1 1 1 1 1, , , ,... ,...
2 4 8 16 2n 2
1 1 1 1 6, , , ,... ,...
2 4 8 16 1 6n n n
Although the first three terms are the same, these are different sequences
We can only write an apparent nth term.
There may be others
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Example 3.
Write an expression for the apparent th term of each sequence.
a. 1, 3, 5, 7,... b. 2, 5, 10, 17,...nn a
:1 2 3 4 ...n n
:1 3 5 7 ... nterms a
apparent pattern: 2 1na n
:1 2 3 4 ...n n
: 2 5 10 17 ... nterms a
apparent pattern:2 1na n
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Example 4.
Write an expression for the apparent th term of the sequence:
2 3 4 5, , , ,...
1 2 3 4
nn a
1n
na
n
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Some sequences are defined recursively.
To define a sequence recursively, you need to be given one or more of the first few terms.
0 1 2 3 1 2
Example 5.
The Fibonacci sequence is defined recursively as follows:
1 1 2 3 2k k ka a a a a a a where k
Write the first five terms of this sequence.
0 1 2 31 1 2 3a a a a
4 3 2 3 2 5a a a 5 4 3 5 3 8a a a
The subscripts of the sequence make up the domain of the sequence and they identify the location of a term within the sequence.
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Factorial Notation
If is a positive integer, factorial is defined as
! 1 2 3 4 ... 1
n n
n n n
zero factorial is defined as 0! = 1
Factorials follow the same rules for order of operations as exponents.
2n! = 2(n!) = 2 1 2 3 4 ... n
2 ! 1 2 3 4 ... 2n n
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Example 6.
2Write the first five terms of the sequence given by .
!Begin with 0
n
na nn
0
0
2 11
0! 1a
1
1
2 22
1! 1a
2
2
2 42
2! 2a
3
3
2 8 4
3! 6 3a
4
4
2 16 2
4! 24 3a
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Example 7.
Evaluate each factorial expression.
8! 2!6! !a. b. c.
2!6! 3!5! 1 !
2 2 ! 2 1 !2 !d. e. f.
2 4 ! ! 2 !
n
n
n nn
n n n
8! 8 7 6! 8 7 56a. 28
2!6! 2!6! 2 2
2!6 5! 6
b. 23 2!5! 3
1 2 3 ... 1!c.
1 ! 1 2 3 ... 1
n nnn
n n
2
2 2 ! 1 2 3 2 2 1 1d.
2 4 ! 1 2 3 2 2 2 3 2 4 2 3 2 4 4 14 12
n n
n n n n n n n n
2 !e. =2
!
n
n 2 1 ! 1 2 3 2 2 1
f. 2 12 ! 1 2 3 2
n n nn
n n
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There is a convenient notation for the sum of the terms of a finite sequence. It is called summation notation or sigma notation because it involves the use of the uppercase Greek letter sigma.
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5 6 82
1 3 0
Example 8.
1a. 3 b. (1 ) c.
!i k n
i kn
5
1
a. 3 3 1 3 2 3 3 3 4 3 5i
i
3 1 2 3 4 5 3 15 45
6
2 2 2 2 2
3
b. (1 ) 1 3 1 4 1 5 1 6k
k
8
0
1 1 1 1 1 1 1 1 1 1c. 2.71828
! 1 1 2 6 24 120 720 5040 40,320n n
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Properties of Sums
1
1. , is a constantn
i
c cn c
1 1
2. , is a constantn n
i ii i
ca c a c
1 1 1
3.n n n
i i i ii i i
a b a b
1 1 1
4.n n n
i i i ii i i
a b a b
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SeriesMany applications involve the sum of the terms of a finite or an infinite sequence. Such a sum is called a series.
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1
Example 9.
3For the series , find a the third partial sum and b the sum.
10i
3
1 2 31
3 3 3 3a. .3 .03 .003 0.333
10 10 10 10ii
1 2 3 4 51
3 3 3 3 3 3 1b. .3 .03 .003 .0003 .00003 0.33333
10 10 10 10 10 10 3ii
Notice that the sum of an infinite series can be a finite number.
Variations in the upper and lower limits of summation can produce quite different-looking summation notation for the same sum.
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Sequences have many applications in situations that involve a recognizable pattern.
2
Example 10.
From 1970 to 2001, the resident population of the United States can be approximated by the
model 205.7 1.78 0.025 , 0,1,...,31 where is the population in millions
and represents thn na n n n a
n
e year, with 0 corresponding to 1970.
Find the last five terms of this finite sequence.
n
2
27 205.7 1.78 27 0.025 27 272.0a
2
28 205.7 1.78 28 0.025 28 275.1a
2
29 205.7 1.78 29 0.025 29 278.3a
2
30 205.7 1.78 30 0.025 30 281.6a 2
31 205.7 1.78 31 0.025 31 284.9a