Geometric Sequences A geometric sequence is a sequence where each # in the seq. is multiplied by a...
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Transcript of Geometric Sequences A geometric sequence is a sequence where each # in the seq. is multiplied by a...
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
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
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 + . . . +256
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?
12+
23+
89+
3227
+. . . .
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
Writing a repeating decimal as a fraction
39.0
Writing a repeating decimal as a fraction
127.0
A different kind of sequence. . .
(𝑥+𝑦 )𝑛
Expand
¿
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 1First 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