Download What is a geometric sequence?

What is a geometric sequence?
A geometric sequence is a pattern of numbers where we keep
multiplying by the same number to obtain the next term.
The number we multiply by is called the common ratio.
2
4
8
16
32
64
128
256
1024
2
This sequence has a common ratio of ...
The next three terms are ...
512
2048, 4096, 8192
These numbers are actually the binary system, used in
computing (e.g. memory capacity, download rates, etc).
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What is a geometric sequence?
Defining a geometric sequence
Decide which of the following sequences are geometric, giving the
common ratio where appropriate:
To define a geometric sequence, we must give the common ratio
and the first term.
2
5
4
12
1
1
2
1
5
99
3
6
9
4.4
8
11
36
8.8
14
108
3
✘
...
✔ common ratio = 3
...
5
8
17.6
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a
✘
...
✘
...
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For a geometric sequence with first term a and common ratio r,
the sequence becomes:
15
35.2
ar3
ar4
...
un = arn – 1
✔ common ratio = 2
...
ar2
So for the nth term (given as un), the index on r is one less than n:
✘
...
ar
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Example 1: Finding terms
Example 2: finding r and a
un = arn – 1
un = arn – 1
The first term of a geometric sequence is 5 and the common ratio
is -2. Find the 2nd, 5th and 10th terms.
2nd term:
ar n 1
 5   2
 10
5th term:
ar n 1
 5   2
 80
10th term:
ar n 1
 5   2 
 2560
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21
5 1
10 1
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The fifth term of a geometric sequence is 48 and the sixth term
is -96. Find the common ratio and first term.
Substitute into one of the
terms to find a.
You can divide consecutive
terms to find the common
ratio.
ar 5
r
ar 4
5
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a  2  48
48
a
16
4
 96
 2
48
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3
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What is a geometric series?
Proof ...
A geometric series is the sum of the terms in a geometric
sequence.
e.g.
Geometric sequence:
2
Geometric series:
2 + 4 + 8 + 16 + 32 + ....
In general terms:
4
8
16
S n  a  ar  ar  ...  ar
2
32
Sn 
....
a 1  r n 
1 r
S n  a  ar  ar 2  ...  ar n 1
a 1  r n 
1 r
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n
S n  rS n  a  ar
S n 1  r   a 1  r n 
Sn 
(given in the exam’s
formulae book)
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Example 4: summing the first n terms
Sn 
The first term of a geometric sequence is 3 and the common
ratio is -2. Find the sum of the first 5 terms.
3 1   2
S5 
1   2 

5
a 1  r n 
1 r

ar
r
ar
18

6
3
a 3   6
a2
 33
1
a 1  r n 
1 r
9
The 2nd term of a geometric sequence is 6 and
the 3rd is 18. Find the sum of the first 5 terms,
and hence the sum of the 6th to 10th terms.
2. Find S5
we only need these
terms
Sn 
a 1  r n 
1 r
2
ar
ar
18

6
r
3
a 3   6
a2
1
S6 to 10 = S10 – S5
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3. Find S6 to 10
21  35 
1 3
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The 2nd term of a geometric sequence is 6 and
the 3rd is 18. Find the sum of the first 5 terms,
and hence the sum of the 6th to 10th terms.
Split the question into manageable chunks:
S10 = u1 + u2 + u3 + u4 + u5 + u6 + u7 + u8 + u9 + u10
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S5 
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1. Find a and r
3. Find S6 to 10
we need to subtract
these terms
2. Find S5
Exam question
Split the question into manageable chunks:
1. Find a and r
The 2nd term of a geometric sequence is 6 and
the 3rd is 18. Find the sum of the first 5 terms,
and hence the sum of the 6th to 10th terms.
 242
Exam question
Sn 
8
a 1  r n 
Sn 
1 r
2
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Split the question into manageable chunks:
1. Find a and r
31  32
1 2
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a1  r n 
1 r
Exam question
a 1  r n 
Sn 
1 r

rSn  ar  ar 2  ar 3  ...  ar n
Multiply Sn by r:
n 1
We can find the sum of the first n terms using the formula:
Sn 
This complicated-looking formula can be
derived quite simply:
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2. Find S5
Sn 
a 1  r
1 r
S5 
21  35 
1 3
n

 242
3. Find S6 to 10
S10 
21  310 
1 3
 59048
S10  S5  59048  242
 58806
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Consider ...
What will happen to the terms in a geometric sequence if r is
negative?
The terms will alternate between positive and negative.
What will happen to the terms in a geometric sequence if r = 1?
All the terms will be the same, so the sequence would not
be classed as geometric.
What will happen to the terms in a geometric sequence if -1 < r < 1?
rn will tend to zero, so the terms will also tend to zero.
What will happen to a geometric series if -1 < r < 1?
The terms will eventually get so small, they will be negligible.
So the series will converge on a particular number.
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