Download INTRO TO SEQUENCES AND SERIES

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A sequence is a function whose domain is a set of
consecutive integers. If a domain is not specified, it is
understood that the domain starts with 1. (A list of
terms with some pattern)
The values in the range are called the terms of the
sequence.
A finite sequence has a limited number of terms.
An infinite sequence continues without stopping.
A sequence can be specified by an equation, or rule.
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When the terms of a sequence are added together,
the resulting expression is a series.
Summation notation or sigma notation, is used
to write a series.
For example, in the series 4 2i 2i is
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i 1
the index of summation, 1 is the lower limit of
summation and 4 is the upper limit of
summation.
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A factorial is the number and each number smaller
than it multiplied together. Its notation is !
For example 5! = 5 * 4 * 3 * 2 * 1 = 120
On the calculator, factorial is found under math, PRB,
option 4
(-1) raised to an even power = 1 and (-1) raised to an
odd power is -1
For example (-1)6 = 1 and (-1)9 = -1
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Write the first four terms of these sequences
7, 11,15, 19
1. an  4n  3
n!
n
3
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2.
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n
a
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(

1)
(3n  2)
3. n
1/3, 2/9, 6/27, 24/81
-1, 4, -7, 10
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Find the pattern, write a rule for the nth term of the
sequence, and then write the next term
12, 15, 18, 21, 24, 27, 30
Look for a pattern
This is going up by 3 each time, this will get multiplied by n, then figure out
the 0th term
12-3 = 9
an = 3n + 9
Pattern
Next term = 33
0th term
34, 27, 20, 13, 6, -1
Look for a pattern
This is going down by 7 each time, this will get multiplied by n,
then figure out the 0th term
34+7 = 41
an = -7n + 41
Pattern
Next term = -8
0th term
-5, -1, 3, 7, 11
Look for a pattern
This is going up by 4 each time, this will get multiplied by n,
then figure out the 0th term
-5-4=-9
an = 4n - 9
Pattern
Next term = 15
0th term
2 3 4 5 6
, , , ,
3 4 5 6 7
Look for a pattern in the numerator and the denominator separately
an = n  1
n2
1, 4, 9, 16, 25, 36, 49
Term # 1, 2, 3, 4,
5,
6,
7
What are you doing to the term # (domain) in order to get the term (range)?
Squaring it!
an = n2
1 1 1
1, , , ,...
3 5 7
Nothing is happening to numerator so it will not have n in it
an =
1
2n  1
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Σ (greek letter sigma) means to add
Plug in all numbers up until this one
4
Means add
a
n 1
It is not important
what this letter is
n
Sequence to plug in to
1st number to plug into ak
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Evaluate
5
 2i  4
i 3
2(3) + 4
36
+
2(4) + 4
+
2(5) + 4
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Evaluate
6
k
2
k 0
0 + 1 + 4 + 9 + 16 + 25 + 36
91
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Evaluate
6

j 4
3
j2
3/6 + 3/7 + 3/8
73/56
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Figure out the pattern and write formula as before
Figure out which numbers are getting plugged in, use
these as the beginning and end numbers in sigma
notation
5 5 5
5
   ..... 
2 3 4
16
5
n 1
1st number that gets plugged in is 1 (to get 2 in the denominator)
last number that gets plugged in is 15 (to get 16 in the denominator)
15
5

n 1 n  1
1 1 1 1
1
    ..... 
1 4 9 16
400
1
n2
1st number that gets plugged in is 1 (to get 1 in the denominator)
last number that gets plugged in is 20 (to get 400 in the denominator)
20
1

2
n
n 1
Sometimes a series is infinite which means the upper
limit is  .
A partial sum is just the sum of the first few terms of an
infinite series. The problem will tell you how many
terms to add.
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Find the
6th

1
partial sum of  n
n 1 10
1
1
1
1
1
1
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101 102 103 104 105 106
1
9
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List math sum(
List ops seq(
Sum(seq(formula, variable, lower limit, upper limit))
Examples:
8
1.  3k  2
k 4
4
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2.  i
i 2
8
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3.  i
i 1
2
i4
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Set mode to seq
Put sequence in y=
Press graph
You may need to change the viewing window to see
enough point
You can also press table (2nd graph) to see the (x,y)
points
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Graph the first 6 terms
an  2n  1
ak  2k
ai  i 2  2