Geometric Sequences
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Arithmetic Sequence ­ there is a common "difference" between each term
= adding the same value between successive terms
*common difference is denoted by d
Geometric Sequence ­ there is a common "ratio" between each term
= multiplying the same value between successive terms
*common ratio is denoted by r
A Geometric Sequence is a sequence in which the same number is multiplied to get from one term to the next. This common ratio is called r.
Find the next 3 terms in the geometric sequence
3, 6, 12, 24, . . . 5, 15, 45, 135, . . . 32, ­16, 8, ­4, . . .
16, 24, 36, 54, . . .
30 Find the next term in geometric sequence: 6, ­12, 24, ­48, 96, . . .
31 Find the next term in geometric sequence: 64, 16, 4, 1, . . .
32 Find the next term in geometric sequence: 6, 15, 37.5, 93.75, . . .
33 Is the following sequence geometric? 48, 24, 12, 8, 4, 2, 1
Geometric Sequences
Geometric Sequences can be described by giving the first term, a , and the common ratio, r.
1
Examples: Find the first five terms of the geometric sequence described.
1) a = 6 and r = 3
1 2) a = 8 and r = ­.5
1
3) a = ­24 and r = 1.5
1
4) a = 12 and r = /
1
2
3
34 Find the first four terms of the geometric sequence described: a = 6 and r = 4.
1
A 6, 24, 96, 384
B
4, 24, 144, 864
C 6, 10, 14, 18
D 4, 10, 16, 22
35 Find the first four terms of the geometric sequence described: a = 12 and r = ­ / .
1
A 12, ­6, 3, ­.75
B
12, ­6, 3, ­1.5
C 6, ­3, 1.5, ­.75
D ­6, 3, ­1.5, .75
1
2
36 Find the first four terms of the geometric sequence described: a = 7 and r = ­2.
1
A 14, 28, 56, 112
B
­14, 28, ­56, 112
C 7, ­14, 28, ­56
D ­7, 14, ­28, 56
Geometric Sequences
Consider the sequence: 3, 6, 12, 24, 48, 96, . . .
To find the seventh term, just multiply the sixth term by 2.
But what if I want to find the 20 th term?
Look for a pattern:
a1
3
a2 6 = 3(2)
a3 12 = 3(4) = 3(2)2 a4 24 = 3(8) = 3(2)3 a5 48 = 3(16) = 3(2)4
a6 96 = 3(32) = 3(2)5 a7 192 = 3(64) = 3(2)6 Do you see a pattern?
Geometric Sequences
Geometric Sequences
Find the indicated term.
Example: a20 given a1 =3 and r = 2.
Example: a10 for 2187, 729, 243, 81
Geometric Sequences
Example: Find r if a = .2 and a = 625
6
1
Example: Find n if a1 = 6, an = 98,304 and r = 4.
37 Find a in a geometric sequence where
a = 5 and r = 3.
12
1
38 Find a in a geometric sequence where
a = 7 and r = ­2.
10
1
39 Find a in a geometric sequence where
a = 10 and r = ­ / .
7
1
1
2
40 Find r of a geometric sequence where
a = 3 and a =59049.
1
10
41 Find n of a geometric sequence where
a = 72, r = .5, and a = 2.25
1
n Geometric Sequences
Find the missing term in the geometric sequence
3, 9, 27,___
5, 1, 1/5 , ____
___, ­10, 50, ­250
­2, ___, ­32
Notice for the last example, r was multiplied to ­2 to get to ___,
and r was again multiplied to the ____ to get to ­32.
OR
So the blank can be either 8 or ­8.
Geometric Sequences
Find the missing terms in the geometric sequence.
5, ___, ___, 40
­54, ___, ___, 16
4, ___, ___, ___, 364
144, ___, ___, ___, ___, 4.5
42 What number(s) fill in the blanks of the geometric sequence? ___, 14, 98, 686
43 What number(s) fill in the blanks of the geometric sequence? 5, ___, 80
44What number(s) fill in the blanks of the geometric sequence: 4, ___, ___, ­500