A geometric sequence or progression, is a sequence where each term is calculated by multiplying the previous term by a fixed number.
Notation
This fixed number is called the common ratio, r.
The common ratio can be positive or negative, an integer or a fraction.
The common ratio can be calculated by dividing any term by the one before it.
e.g. Common ratio = t n+1 ÷ t n
The first term of a geometric sequence is shown by the variable a.
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Geometric Sequence
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First term, a
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Common ratio, r
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2, 6, 18, 54, ...
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2
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6 ÷ 2 = 3 |
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20, 10, 5, 2.5, ...
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20
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10 ÷ 20 = 0.5
5 ÷ 10 = 0.5 etc. |
General Term, tn
A geometric sequence can be written:
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First term
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Second term
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Third term
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Fourth term
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General term (n th term)
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t 1
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t 2
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t 3
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t 4
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...
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t n
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a
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ar
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ar2
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ar3
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...
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ar n − 1
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Example 1
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What is the common ratio of the geometric sequence: 20, 10, 5, 2.5,, ... |
Common ratio = t n+1 ÷ t n Second term ÷ first term = t2 ÷ t1 = 10 ÷ 20 = 0.5 Check: Third term ÷ second term = 5 ÷ 10 = 0.5 The common ratio is 0.5 |
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Example 2
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Find the 8th term of the geometric sequence: 3, 9, 27, 81, ... |
Common ratio, r = 9 ÷ 3 = 3 using tn = ar n − 1 t8 = 3 x 3 8- 1 The 8th term is 6561 |
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Example 3
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Which term of the sequence 2, 4, 8, 16, ... would be equal to 1024? |
Common ratio, r = 2 Using tn = ar n-1 1024 =2 x 2 n-1 1024 is the 10th term. |
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Example 4
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The third term of a geometric sequence is 8 and the fifthterm is 32. Find the first term, a, and the common ratio, r, and thus list the first four terms of the sequence. |
t 3 = 8 using tn = ar n-1 32 = ar 5 − 1 32 = ar 4 4 = r2 8 = a x 22 The sequence is 2, 4, 8, 16, ... or 2, -4, 8, -16... |
