4/27/2012
Warm Up
Sequences and Series
Warm Up #2
Sequence
• A sequence is a function whose
domain is a set of consecutive
integers
Terms
• The terms are the values or numbers
in the sequence
Finite Sequence
A finite sequence has a limited number
of terms
Infinite Sequence
An infinite sequences continues
without stopping.
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4/27/2012
Write terms of a
sequence
Series
When terms of a sequence are added
together, the resulting expression
is a series.
Summation notation or Sigma
notation is used to write a series.
• Recursive formula: uses the term
before to find the next term
• Discrete formula: plug directly into
the given rule to find each term
k
 rule
n =1
Recursive formula
a1 = 3
an = an −1 + 6
Write the first 4 terms of the sequence
a1 = 3
a2 = a2−1 + 6
a2 = 3 + 6 = 9
a3 = a3−1 + 6
a3 = 9 + 6 = 15
a4 = a4−1 + 6
a4 = 15 + 6 = 21
Find a term
• Recursive formula, must know the
terms before
• Discrete formula, just substitute
into the formula to get the term you
need
Discrete Formula
•
•
•
•
•
•
•
an = 2n
n = number of the term
Write the first 4 terms
a1 = 2(1)=2
a2 = 2(2) = 4
a3 = 2(3) = 6
a4 = 2(4) =8
How do you know what type
of formula you are given?
• Recursive formulas will have an-1 in
the formula and you will be given a
beginning term
• Discrete formula will just have n in
the formula.
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4/27/2012
Find the rule given the
sequence
A Famous Infinite Sequence
•
•
•
•
•
•
•
1,1,2,3,5,8,…
Write the series using
summation notation
• 7+14+21+…+77
• Rule for the sequence is 7n
• Summation notation
11
 7n
n =1
Find the sum of the
series
Write the series then add
7, 11, 15, 19, ……
Can you find a pattern?
Added 4 each time
Recursive
a1=7
an= an-1+4
Discrete
an =4n+3
Write the series using
summation notation
• -4 -8 -12 -16- ………
• Infinite series
• Rule of sequence is -4n
∞
 −4n
n =1
Special sequences
• Arithmetic sequence- common difference
between terms
5
n
2
n =1
1+4+9+16+25 = 55
• Geometric sequence – common ratio
(product) between terms.
WE WILL STUDY ARITHMETIC
NEXT YEAR YOU WILL STUDY GEOMETRIC
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4/27/2012
Arithmetic Sequence
• A sequence in which the difference
between consecutive terms is
constant
• Common difference is the constant
difference between the terms
denoted by d.
Arithmetic Series
• The expression formed by adding the
terms of an arithmetic sequence this
is denoted by Sn
Write a rule for the nth
term of the arithmetic
sequence
Rule for an Arithmetic
Sequence
• The nth term of an arithmetic
sequence with first term a1 and
common difference d is given by:
• an=a1 + (n-1)d
• a1 = the first term
• d = common difference
• n = number of term you want to find
You could use….
• Since the arithmetic sequence has a common
difference if you graph the sequence it is linear.
Therefore, You could find the rule for the
arithmetic sequence by finding the equation of
the line…
• 1 2 3 4 ……
• 7,10,13,16,….
• Slope is 3
• plug into y = mx + b solve for b 7=1(3)+b b = 4
• Rule is 3n+4
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•
•
•
7,10,13,16,…..
an=a1 + (n-1)d
a1 = 7 d = 3
an = 7 +(n-1)3
= 7 +3n – 3
= 4 + 3n
Now you could find any term of the sequence by
plugging into n .
Find the 8th term of the
sequence
• 8,9,10,11,12,…..
• an=a1 + (n-1)d
•
= 8 + (8-1)1
•
= 8 + (7)
•
= 15
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4/27/2012
The sum of a finite
arithmetic series
You could
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Find the rule: plug in
8,9,10,11,12
Slope is 1 8 = 1(1)+b b = 7
Rule is n+7
8th term is 8 + 7 = 15
n
( a1 + an )
2
Sum of the first n terms
Sn =
Sn
n is the number of terms
a1 is the first term
an is the nth term (last term)
Find the sum of the first
30 terms of the series
2+4+6+8+…..
Sn =
n
( a1 + an )
2
s30 =
need to find a30
30
( 2 + a30 )
2
s30 = 15(62)
Use the formula to find a certain term
•an = a1+ (n-1)d
s30 = 930
= 2+(30-1)2
60 is the 30th term
Find the value of n
(−3 + ) = 18
You know the sum is 18, so use the sum formula:
2
+
= 18
= −2;
= −3 +
Find the sum of the
arithmetic series
4
1.I can write
the series out
and add:
 n +1
2+3+4+5
n =1 = 14
OR
2. I can use the formula:
4((2+5)/2) = 14
homework
p. 135 1-25 ODD
p. 141 1-9 ; 19-27;
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