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A geometric sequence is a sequence where each term is found by multiplying or dividing by the same value from one term to the next. The number we multiply by is called the "common ratio". Dividing by 3 is the same as multiplying by 13, so then the common ratio is 13.
A sequence is a set of numbers that follow a pattern. We call each number in the sequence a term.
For example, the following two sequences are examples of geometric sequences.
2, 4, 8, 16, 32, 64, ...
As you can see, each term is found by multiplying the previous term by 2, the common ratio. Notice that we have to add 2 to the first term to get the second term, but we have to add 4 to the second term to get 8. This shows indeed that this sequence is not created by adding or subtracting a common difference.
243, 81, 27, 9, 3, 1, ...
This time, to find each term, we divide the previous term by 3. The common ratio is 13.
Looking for a number that is added each time. In a geometric sequence the terms are multiplied by the same number, so the gaps between terms change: 2, 4, 8, 16 has gaps 2, 4 and 8. To find the common ratio, divide a term by the term before it: 8 ÷ 4 = 2.
What is the common ratio of 5, 15, 45, 135, ...?
3. Divide a term by the one before it: 15 ÷ 5 = 3 and 45 ÷ 15 = 3.
Many geometric sequences can be modeled with an exponential function. An exponential function is a function of the form an where a > 0, a ≠ 1 and n is a variable.
Here is a trick or "recipe per se" to quickly get an exponential function!
1) Let us try to model 2, 4, 8, 16, 32, 64, ...
Let n represent any term number in the sequence.
Observe that the terms of the sequence can be written as 21, 22, 23, ...
We can therefore model the sequence with this exponential function: 2n
Check to see if the exponential function works:
When n = 1, which represents the first term, we get 21 = 2
When n = 2, which represents the second term, we get 22 = 4
When n = 3, which represents the third term, we get 23 = 8
The exponential function works!
2) Let us try to model 243, 81, 27, 9, 3, 1, ...
Let n represent any term number in the sequence.
Observe that the terms of the sequence can be written as 35, 34, 33, ...
We just have to model the exponents: 5, 4, 3, ...
The process will be briefly explained here. For detailed explanation, see arithmetic sequence.
The number we add to each term is −1.
The number that comes right before 5 in the sequence is 6.
5, 4, 3, ... can be modeled with the following algebraic expression: −1 × n + 6
We can therefore model 243, 81, 27, 9, 3, 1, ... with the exponential function below: 3−n + 6
Check to see if the exponential function works:
When n = 1, which represents the first term, we get 3−1 + 6 = 35 = 243
When n = 2, which represents the second term, we get 3−2 + 6 = 34 = 81
When n = 3, which represents the third term, we get 3−3 + 6 = 33 = 27
The exponential function works!
Which exponential function models 5, 25, 125, 625, ...?
5n. The terms are 51, 52, 53, 54. Check: n = 3 gives 53 = 125.
1) 2n is the exponential function for 2, 4, 8, 16, 32, 64, ...
Let us try to rewrite 2n by making the first term appear in the exponential function.
2n = 2 × 2n × 2−1 (since 2 × 2−1 = 20 = 1 and 1 × 2n = 2n)
2n = 2 × 2n − 1
2 is the number we multiply each term by
2 is the first term
n is the term number
2) 3−n + 6 is the exponential function for 243, 81, 27, 9, 3, 1, ...
Let us try to rewrite 3−n + 6 by making the first term, also equal to 35, appear in the exponential function.
3−n + 6 = 35 × 3−n + 6 × 3−5 (since 35 × 3−5 = 30 = 1 and 1 × 3−n + 6 = 3−n + 6)
3−n + 6 = 35 × 3−n + 1
3−n + 6 = 35 × 3−1(n − 1)
3−n + 6 = 35 × (13)n − 1 (3−1 = 13)
13 is the number we multiply each term by and it means the same as dividing each term by 3.
35 is the first term
n is the term number
In general,
Let r be the number we multiply each time or the common ratio.
Let a1 be the first term
Let n be the term number
Let an be the nth term.
Then,
The exponent is n − 1, not n. You start at the first term a1 and multiply by r one time less than n. For 2, 4, 8, ..., the 3rd term is 2 × 23 − 1 = 2 × 4 = 8.
Answer: a6 = 96
Find the 5th term of 2, 6, 18, 54, ...
162. r = 3 and a1 = 2, so a5 = 2 × 34 = 2 × 81 = 162.
Are the given sequences geometric? If so, find the 9th term.
a. 20, 5, 1, 0.25, ...
20, 5, 1, 0.25, ... is not a geometric sequence since the number we divide each term by is not always the same.
b. 12, 3, 0.75, 0.1875, ...
12, 3, 0.75, 0.1875, ... is a geometric sequence since the number we always divide each term by is 4.
Then, an = a1 × rn − 1
r = 14 = 0.25
n = 9
a1 = 12
a9 = 12 × (0.25)9 − 1
a9 = 12 × (0.25)8
a9 = 12 × 0.0000152587890625
a9 = 0.00018310546875
Is 2, 6, 12, 20, ... a geometric sequence?
No. 6 ÷ 2 = 3 but 12 ÷ 6 = 2, so the terms are not multiplied by the same number.
Now check what you learned. Every answer is explained, and each time you play you get new questions.
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