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...
![Page 1: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/1.jpg)
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
![Page 2: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/2.jpg)
Geometric Sequences
General Formula:1
1 n
n raa
![Page 3: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/3.jpg)
Geometric Sequences
Find the 11th term of the geo. Sequence listed below
64, -32, 16, -8,…a11
11
nn raa
![Page 4: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/4.jpg)
Geometric Sequences
Find the 6th term of the geo. sequence listed below
3, -15, 75, ..a6
11
nn raa
![Page 5: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/5.jpg)
Geometric Sequences
Write an equation for the nth term
3, 12, 48, 192...an
11
nn raa
![Page 6: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/6.jpg)
Geometric Sequences
Find the 10th term of the sequence if
a4=108
r=3
11
nn raa
![Page 7: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/7.jpg)
Geometric Sequences
Find the 7th term of the sequence if
a3=96
r=2
11
nn raa
![Page 8: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/8.jpg)
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
![Page 9: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/9.jpg)
Geometric Means
Find 5 geometric means between ½ and 1/1458
![Page 10: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/10.jpg)
Geometric Series
A series that is associated with a geometric sequence
r
raS
n
n
1
11
![Page 11: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/11.jpg)
Geometric Series
Find the sum of the first 6 terms of the geometric series
3 + 6 + 12 + 24 +…
r
raS
n
n
1
11
![Page 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/12.jpg)
Geometric Series
Find the first term of the series if the S8=39,360 and r=3
r
raS
n
n
1
11
![Page 13: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/13.jpg)
Geometric Series
Find the sum of the first 8 terms of
1+x+x2+x3+…
r
raS
n
n
1
11
![Page 14: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/14.jpg)
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
![Page 15: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/15.jpg)
Sigma Notation
8
5
43j
j
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Sigma Notation
10
3
12k
k
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Sigma Notation
5
1
1
2
1
n
n
![Page 18: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/18.jpg)
Write in sigma notation
1 + 3 + 5 + 7
![Page 19: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/19.jpg)
Write in sigma notation
2 + 4 + 6 + 8 + 10
![Page 20: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/20.jpg)
Write in sigma notation
3 + 6 + 12 + 24 + 48
![Page 21: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/21.jpg)
Write in sigma notation
-3 + 9 + -27 + 81 + -243
![Page 22: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/22.jpg)
Write in sigma notation
-2 + 4 + -8 + . . . +256
![Page 23: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/23.jpg)
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
![Page 24: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/24.jpg)
Does the geom. series have a sum?
...81
1
27
1
9
1
3
11
![Page 25: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/25.jpg)
Does the geom. series have a sum?
...625
3
125
3
25
3
5
33
![Page 26: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/26.jpg)
Does the sum of each term approach some limit?
12+
23+
89+
3227
+. . . .
![Page 27: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/27.jpg)
To find the sum of an infinite series
Make sure a limit exists first
r
aSn
1
1
![Page 28: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/28.jpg)
An infinite series in sigma notation—find the sum
1
1 5
124
n
n
![Page 29: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/29.jpg)
An infinite series in sigma notation—find the sum
1
1 2
15
n
n
![Page 30: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/30.jpg)
Writing a repeating decimal as a fraction
39.0
![Page 31: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/31.jpg)
Writing a repeating decimal as a fraction
127.0
![Page 32: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/32.jpg)
A different kind of sequence. . .
(𝑥+𝑦 )𝑛
![Page 33: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/33.jpg)
Expand
¿
![Page 34: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/34.jpg)
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
![Page 35: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/35.jpg)
Expand using Pascal’s Triangle
52 yx
![Page 36: 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.](https://reader036.fdocuments.us/reader036/viewer/2022070418/5697c02b1a28abf838cd8bd3/html5/thumbnails/36.jpg)
Expand using Pascal’s Triangle
423 yx