u sing and w riting s equences the numbers in sequences are called terms. you can think of a...
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USING AND WRITING SEQUENCES
The numbers in sequences are called terms.
You can think of a sequence as a function whose domainis a set of consecutive integers. If a domain is notspecified, it is understood that the domain starts with 1.
The domain gives the relative position of each term.
1 2 3 4 5 DOMAIN:
3 6 9 12 15RANGE:The range gives the terms of the sequence.
This is a finite sequence having the rule
an = 3n,where an represents the nth term of the sequence.
USING AND WRITING SEQUENCES
n
an
Writing Terms of Sequences
Write the first six terms of the sequence an = 2n + 3.
SOLUTION
a 1 = 2(1) + 3 = 5 1st term
2nd term
3rd term
4th term
6th term
a 2 = 2(2) + 3 = 7
a 3 = 2(3) + 3 = 9
a 4 = 2(4) + 3 = 11
a 5 = 2(5) + 3 = 13
a 6 = 2(6) + 3 = 15
5th term
Writing Terms of Sequences
Write the first six terms of the sequence f (n) = (–2)
n – 1 .
SOLUTION
f (1) = (–2) 1 – 1 = 1 1st term
2nd term
3rd term
4th term
6th term
f (2) = (–2) 2 – 1 = –2
f (3) = (–2) 3 – 1 = 4
f (4) = (–2) 4 – 1 = – 8
f (5) = (–2) 5 – 1 = 16
f (6) = (–2) 6 – 1 = – 32
5th term
Writing Rules for Sequences
If the terms of a sequence have a recognizable pattern, then you may be able to write a rule for the n th term of the sequence.
Describe the pattern, write the next term, and write a rule for the n th term of the sequence
____ – , , – , , ….13
19
127
181
_ _
1 3
, 1 9
, 1 27
, 1 81
Writing Rules for Sequences
SOLUTION
1 2 3 4n
terms 1243
5
13
4
13
1, 1
3
2, 1
3
3, 1
3
5rewriteterms
13
A rule for the nth term is an =
n
2 6 12 20
Writing Rules for Sequences
SOLUTION
A rule for the nth term is f (n) = n (n+1).
terms
5(5 +1)
Describe the pattern, write the next term, and write a rule for the n th term of the sequence.
2, 6, 12 , 20,….
5
30
1 2 3 4
rewriteterms 1(1 +1) 2(2 +1) 3(3 +1) 4(4 +1)
n
Graphing a Sequence
You can graph a sequence by letting the horizontal axisrepresent the position numbers (the domain) and the vertical axis represent the terms (the range).
Graphing a Sequence
You work in the producedepartment of a grocery storeand are stacking oranges in the shape of square pyramid with ten layers.
• Write a rule for the number of oranges in each layer.
• Graph the sequence.
Graphing a Sequence
SOLUTION
The diagram below shows the first three layers of the
stack. Let an represent the number of oranges in layer n.
n 1 2 3
an 1 = 1 2 4 = 2 2 9 = 3
2
From the diagram, you can see that an = n
2
Graphing a Sequence
Plot the points (1, 1), (2, 4), (3, 9), . . . , (10, 100).
an = n2
USING SERIES
. . .
3 + 6 + 9 + 12 + 15 = ∑ 3i5
i = 1
FINITE SEQUENCE
FINITE SERIES
3, 6, 9, 12, 15
3 + 6 + 9 + 12 + 15
INFINITE SEQUENCE
INFINITE SERIES
3, 6, 9, 12, 15, . . .
3 + 6 + 9 + 12 + 15 + . . .
You can use summation notation to write a series. Forexample, for the finite series shown above, you can write
When the terms of a sequence are added, the resultingexpression is a series. A series can be finite or infinite.
3 + 6 + 9 + 12 + 15 = ∑ 3i5
i = 1
USING SERIES
5
i = 1∑3i
Is read as “the sum from i equals 1 to 5 of 3i.”
index of summation lower limit of summation
upper limit of summation
USING SERIES
Summation notation is also called sigma notation because it uses the uppercase Greek letter sigma, written ∑.
Summation notation for an infinite series is similarto that for a finite series. For example, for the infiniteseries shown earlier, you can write:
3 + 6 + 9 + 12 + 15 + = ∑ 3i∞
i = 1. . .
The infinity symbol, ∞, indicates that the series continues without end.
USING SERIES
The index of summation does not have to be I. Any letter can be used. Also, the index does not have to begin at 1.
Example: Write the series represented by the summation notation . Then find the sum.
= 12 + 12 + 12 + 12 0! 1! 2! 3!
= 12 + 12 + 12 + 12 1 1 2 6
= 32
12n!k=0
3
∑
Writing Series with Summation Notation
Write the series with summation notation.
5 + 10 + 15 + + 100. . .
SOLUTION
Notice that the first term is 5 (1), the second is 5 (2),the third is 5 (3), and the last is 5 (20). So the termsof the series can be written as:
ai = 5i where i = 1, 2, 3, . . . , 20
The summation notation is ∑ 5i.20
i = 1
Writing Series with Summation Notation
Notice that for each term the denominator of the fractionis 1 more than the numerator. So, the terms of the seriescan be written as:
ai = where i = 1, 2, 3, 4 . . . ii + 1
Write the series with summation notation.
SOLUTION
. . .1 2 3 42 3 4 5
+ + + +
∞The summation notation for the series is ∑
i = 1
ii + 1
.
Writing Series with Summation Notation
The sum of the terms of a finite sequence can be foundby simply adding the terms. For sequences with manyterms, however, adding the terms can be tedious. Formulas for finding the sum of the terms of three special types of sequences are shown next.
Writing Series with Summation Notation
FORMULAS FOR SPECIAL SERIESCONCEPT
SUMMARY
n
i = 1∑ 1 = n
∑ i = n (n + 1)
2
n
i = 1
1
2
3
gives the sum of positive integers from 1 to n .
gives the sum of squares of positive integers from 1 to n.
∑ i 2 = n (n + 1)(2 n + 1)
6
n
i = 1
gives the sum of n 1’s .
Using a Formula for a Sum
RETAIL DISPLAYS How many oranges are in a square pyramid 10 layers high?
Using a Formula for a Sum
SOLUTION
You know from the earlier example that the i th term of the series is given by ai = i
2, where i = 1, 2, 3, . . . , 10.
∑10
i = 1i
2 = 12+ 22 + + 102 . . .
10(11)(21)=
6
= 385
There are 385 oranges in the stack.
=6
10(10 + 1)(2 • 10 + 1)