unit 7: sequences and series. sequences a sequence is a list of #s in a particular order if the...
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Unit 7: Sequences and Series
Sequences
A sequence is a list of #s in a particular orderIf the sequence of numbers does not end,
then it is called an infinite sequenceEach # in a sequence is called a termEx. 3, 5, 7, 9….a1=3a2=5
Arithmetic Sequences
A sequence in which each term after the first term is found by adding a constant (called the common difference (d)), to the previous term
Ex. Find the common difference (one term minus the previous term)
55, 49, 43, 37, 31, 25,19
Arithmetic Sequences
Formula for the general pattern for any arithmetic sequence:
differencecommond
termsofn
termtheofvaluea
termntheofvaluea
where
dnaa
st
thn
n
#
1
...
)1(
1
1
Arithmetic Sequences
Write a equation for the nth term of the following sequence
...,35,26,17,8
dnaan )1(1
Write a equation for the nth term of the following sequence
...,2
7,3,
2
5,2
Arithmetic Sequences dnaan )1(1
Arithmetic Sequences
Find a15
,...3,5,7,9
dnaan )1(1
Arithmetic Sequences
In the sequence below, which term has a value of 286?
...,14,10,6,2
dnaan )1(1
Arithmetic Sequences
What is the value of the first term if the 9th and 10th terms are 4 and 2 consecutively?
dnaan )1(1
Arithmetic Sequences
If the 3rd term of an arithmetic sequence is 8 and the 16th term is 47, find a1 and d
dnaan )1(1
Arithmetic Means
Terms between any two non-successive terms of an arithmetic sequence
Find 4 arithmetic means between 16 and 91
dnaan )1(1
Arithmetic Means
Find 1 arithmetic mean between 50 and -120
dnaan )1(1
Results-4A
0123
• Concerns: arithmetic means, notation, 2x problem, more practice
Arithmetic Series
A series is the indicated sum of the terms of a sequence
If the seq. is 18, 22, 26, 30The arith series is 18 + 22 + 26 + 30
Arithmetic Series
To find the sum of an arithmetic series,
nn aan
S 12
Arithmetic Series
Find the sum of the first 100 positive integers.
nn aan
S 12
Arithmetic Series
Find the sum of the first 20 even numbers beginning with 2.
nn aan
S 12
Arithmetic Series
Find the sum of 34+30+26+…+2 nn aa
nS 12
Arithmetic Series
Find the sum of 6+13+20+27…+97 nn aa
nS 12
Geometric Sequences
A geometric sequence is a sequence where each # in the seq. is multiplied by a constant (which is called the common ratio (r))
To find r, divide any term in the sequence by the previous term
Geometric Sequences
General Formula:1
1 n
n raa
Geometric Sequences
Find the 11th term of the geo. Sequence listed below
64, -32, 16, -8,…a11
11
nn raa
Geometric Sequences
Find the 6th term of the geo. sequence listed below
3, -15, 75, ..a6
11
nn raa
Geometric Sequences
Write an equation for the nth term
3, 12, 48, 192...an
11
nn raa
Geometric Sequences
Find the 10th term of the sequence if
a4=108
r=3
11
nn raa
Geometric Sequences
Find the 7th term of the sequence if
a3=96
r=2
11
nn raa
Geometric Means
Geometric means are the missing terms between two non-successive terms in a geo. Sequence
Find 3 geometric means between 2.25 and 576
Geometric Means
Find 5 geometric means between ½ and 1/1458
Geometric Series
A series that is associated with a geometric sequence
r
raS
n
n
1
11
Geometric Series
Find the sum of the first 6 terms of the geometric series
3 + 6 + 12 + 24 +…
r
raS
n
n
1
11
Geometric Series
What is r if the sum of the first 6 terms in a geo series is 11,718 and the first term is 3
Hint: solve by doing an intersection on the graph
r
raS
n
n
1
11
Geometric Series
Find the first term of the series if the S8=39,360 and r=3
r
raS
n
n
1
11
Geometric Series
Find the sum of the first 8 terms of
1+x+x2+x3+…
r
raS
n
n
1
11
Geometric Series
Find the a1 if Sn=165, an=48, r=-2/3
Hint: an=a1.rn-1
So, r.an=a1.rn-1.ran.r=a1.rn (now
substitute into the sum formula)
r
raS
n
n
1
11
r
raaS nn
11
Sigma Notation
More concise (less time consuming) notation for writing out a series
n) case, (in this variable theissummation ofindex the
10" to1 from goesn as3n of sum the"
3
30...12963:10
1
n
n
ex
Sigma Notation
8
5
43j
j
Sigma Notation
10
3
12k
k
Sigma Notation
5
1
1
2
1
n
n
Write in sigma notation
1 + 3 + 5 + 7
Write in sigma notation
2 + 4 + 6 + 8 + 10
Write in sigma notation
3 + 6 + 12 + 24 + 48
Write in sigma notation
-3 + 9 + -27 + 81 + -243
Write in sigma notation
-2 + 4 + -8 + 16 + -32 + 64
Infinite Geometric Series
In an infinite series, Sn approaches some limit as n becomes very large. That limit is defined to be the sum of the series. If an infinite series has a sum, it is said to converge.
A series converges (or has a sum) if and only if lrl < 1
Does the geom. series have a sum?
...81
1
27
1
9
1
3
11
Does the geom. series have a sum?
...625
3
125
3
25
3
5
33
Does the sum of each term approach some limit?
...64
1
32
1
16
1
8
1
4
1
2
1
To find the sum of an infinite series
Make sure a limit exists first
r
aSn
1
1
An infinite series in sigma notation—find the sum
1
1 5
124
n
n
An infinite series in sigma notation—find the sum
1
1 2
15
n
n
Pascal’s Triangle
11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 1*First row is used for anything to the zero
power**used for the coefficients of each term of the
expanded binomial
Expand using Pascal’s Triangle
52 yx
Expand using Pascal’s Triangle
423 yx
Writing a repeating decimal as a fraction
39.0
Writing a repeating decimal as a fraction
127.0