9.1 geometric sequences these are sequences where the ratio of successive terms of a sequence is...
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9.1 GEOMETRIC SEQUENCES
These are sequences where the ratio of successive terms of a sequence is always the same number. This number is called the common ratio.
Do you remember what an arithmetic sequence is?What formula is used?
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An introduction…………
1, 4, 7,10,13
9,1, 7, 15
6.2, 6.6, 7, 7.4
, 3, 6
Arithmetic Sequences
ADDTo get next term
2, 4, 8,16, 32
9, 3,1, 1/ 3
1,1/ 4,1/16,1/ 64
, 2.5 , 6.25
Geometric Sequences
MULTIPLYTo get next term
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Ex: Determine if the sequence is geometric. If so, identify the common ratio
• 1, -6, 36, -216
yes. Common ratio=-6
• 2, 4, 6, 8
no. No common ratio
This is an Arithmetic Sequence with “common difference” of 2
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Important Formula for Geometric Sequence:
an = a1 r n-1
Where:
an is the nth term in the sequence
a1 is the first term
n is the number of the term
r is the common ratio
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Ex: Write the first 4 terms of this sequence with:
First term: a1 = 7
Common ratio = 1/3
an = a1 * r n-1
Now find the first five terms:a1 = 7(1/3) (1-1) = 7a2 = 7(1/3) (2-1) = 7/3a3 = 7(1/3) (3-1) = 7/9a4 = 7(1/3) (4-1) = 7/27a5 = 7(1/3) (5-1) = 7/81
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Geometric Sequence Problem
Find the 19th term in the sequence of 11,33,99,297 . . .
a19 = 11(3)18 =4,261,626,379
Common ratio = 3
a19 = 11 (3) (19-1)
Start with the sequence formula
Find the common ratio between the values.
Plug in known values
Simplify
an = a1 * r n-1
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Let’s try one
Find the 10th term in the sequence of 1, -6, 36, -216 . . .
a10 = 1(-6)9 = -10,077,696
Common ratio = -6
a10 = 1 (-6) (10-1)
Start with the sequence formula
Find the common ratio between the values.
Plug in known values
Simplify
an = a1 * r n-1
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2 2 2 2 r = 2
1 nn ara
Try this to get the 5th term.
a = 1
1621 155 a
1, 2, 4, 8, 16 . . .
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Find the 8th term of 0.4, 0.04. 0.004, . . .
1 nn ara
1.04.0
04.0r
To find the common ratio, take any term and divide it by the term in front
11.04.0 nna
00000004.01.04.0 188 a
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Find the next four terms of –9, -2, 5, …
Arithmetic Sequence
2 9 5 2 7
7 is referred to as the common difference (d)
Common Difference (d) – what we ADD to get next term
Next four terms……12, 19, 26, 33
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Find the next four terms of 0, 7, 14, …
Arithmetic Sequence, d = 7
21, 28, 35, 42
Find the next four terms of x, 2x, 3x, …
Arithmetic Sequence, d = x
4x, 5x, 6x, 7x
Find the next four terms of 5k, -k, -7k, …
Arithmetic Sequence, d = -6k
-13k, -19k, -25k, -32k
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Given an arithmetic sequence with 15 1a 38 and d 3, find a .
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
x
15
38
NA
-3
n 1a a n 1 d
38 x 1 15 3
X = 80
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16 1Find a if a 1.5 and d 0.5 Try this one:
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
1.5
16
x
NA
0.5
n 1a a n 1 d
16 1.5 0.a 16 51
16a 9
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n 1Find n if a 633, a 9, and d 24
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
9
x
633
NA
24
n 1a a n 1 d
633 9 21x 4
633 9 2 244x
X = 27
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1 29Find d if a 6 and a 20
1a First term
na nth term
nS sum of n terms
n number of terms
d common difference
-6
29
20
NA
x
n 1a a n 1 d
120 6 29 x
26 28x
13x
14
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Example 7. An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows?
1 20 1 19d c
1 201 20 19 1 39na a n d a
20
2020 39 10 59 590
2S
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Example 8. A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation.
1 10,000 7500 10,000 7500 2500a d c
1 201 10,000 19 7500 152,500na a n d a
20
2010,000 152,500 10 162,500 1,625,000
2S
So the total sales for the first 2o years is $1,625,000
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Find the next three terms of 2, 3, 9/2, ___, ___, ___
3 – 2 vs. 9/2 – 3… not arithmetic3 9 / 2 3
1.5 geometric r2 3 2
3 3 3 3 3 3
2 2 2
92, 3, , , ,
2
9 9 9
2 2 2 2 2 2
92, 3, , ,
27 81 243
4 8,
2 16
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1 9
1 2If a , r , find a .
2 3
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
1/2
x
9
NA
2/3
n 1n 1a a r
9 11 2
x2 3
8
8
2x
2 3
7
8
2
3 128
6561
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9Find a of 2, 2, 2 2,...
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
x
9
NA
2
2 2 2r 2
22
n 1n 1a a r
9 1
x 2 2
8
x 2 2
x 16 2