2, 4, 8, 16, …
description
Transcript of 2, 4, 8, 16, …
2, 4, 8, 16, … 32Exercise
2, 4, 6, 8, … Exercise
10
1, 3, 9, 27, … 81Exercise
1, , , , … 12
14
18
116
Exercise
1, –2, 4, –8, 16, … –32Exercise
3 6
x 2
12
x 2
24
x 2
48
x 2
96
x 2
Geometric SequenceA geometric sequence is a sequence of numbers whose successive terms differ by a constant multiplier.
Common RatioThe constant multiplier for a geometric sequence is called the common ratio, r.
State whether the sequence 8, 4, 2, 1, … is arithmetic or geometric.
geometric
Example
State whether the sequence –6, –18, –54, –162, … is arithmetic or geometric.
geometric
Example
State whether the sequence 5, 7, 9, 11, … is arithmetic or geometric.
arithmetic
Example
State whether the sequence 5, 10, 20, 40, … is arithmetic or geometric.
geometric
Example
Geometric SequenceTerms differ by a constant factor r.
an = an – 1r
Write the first six terms of the geometric sequence in which a1 = 1 and r = 3.a1 = 1a2 = 1 • 3 = 3a3 = 3 • 3 = 9a4 = 9 • 3 = 27
Example 1
The first six terms of the sequence are
1, 3, 9, 27, 81, and 243.
Write the first six terms of the geometric sequence in which a1 = 1 and r = 3.
Example 1
a5 = 27 • 3 = 81a6 = 81 • 3 = 243
Find the value of a1 for the sequence 2, 6, 18, 54, 162, 486, …
a1 = 2
Example 2
Find the value of r for the sequence 2, 6, 18, 54, 162, 486, …
r = 3
Example 2
Find the value of a3 for the sequence 2, 6, 18, 54, 162, 486, …
a3 = 18
Example 2
Find the value of a8 for the sequence 2, 6, 18, 54, 162, 486, …
= 4,374
a7 = 486 • 3= 1,458
a8 = 1,458 • 3
Example 2
Geometric SequenceTerms differ by a constant factor r.
an = an – 1r
Write the first five terms of the sequence defined bya1 = –4 and an = 3an – 1.a1 = –4a2 = 3(–4) = –12a3 = 3(–12) = –36a4 = 3(–36) = –108a5 = 3(–108) = –324
Example 3
Write the first five terms of the sequence defined bya1 = –4 and an = 3an – 1.
The first five terms of the sequence are
–4, –12, –36, –108, and –324.
Example 3
Write the first four terms of the sequence defined bya1 = 2 and an = 4an – 1.
2, 8, 32, 128
Example
Write the first four terms of the sequence defined bya1 = –3 and an = 2an – 1.
–3, –6, –12, –24
Example
Find the common ratio, r, of the sequence 4, –12, 36, –108.
r = –3
Example
Find the common ratio, r, of the sequence 24, 12, 6, 3.
r =
12
Example
Write the recursive formula for the sequence 729, 243, 81, 27, ...a1 = 729r =
13
an = an – 113
Example 4
3, 6 ,12, 24, 48, 96× 2 to get next term
1Position Term
23456n
33 • 21 = 63 • 22 = 123 • 23 = 243 • 24 = 483 • 25 = 963 • 2n – 1
Explicit FormulaThe explicit formula for a geometric sequence is an = a1r
n –1, in which a1 is the
first term and r is the common ratio.
Write the explicit formula for the sequence –5, –15, –45, –135, –405, ...a1 = –5r = 3 an = –5(3)n – 1
Example 5
Write the explicit formula for the sequence –3, –6, –12, –24, ...
an = –3(2)n – 1
Example
Write the explicit formula for the sequence 12, 6, 3, 1.5, ...
an = 12( )n – 112
Example
A ball bounces three-fourths the height of its fall. If the ball falls 12 ft., how high does it bounce on the first bounce? on the second bounce? on the third bounce?
9 ft.; 6.75 ft.; 5.0625 ft.
Exercise
In the last problem, the height of the bounces forms a geometric sequence. Find the common ratio of this geometric sequence.
r = 0.75
Exercise
If the ball falls 12 ft. and begins bouncing, what is the total distance it has traveled when it hits the ground the third time?
43.5 ft.
Exercise
When will the ball stop bouncing?
Exercise