Before talking about geometric sequence, in math, a sequence is a set of numbers that follow a pattern. We call each number in the sequence a term.
For examples, the following are sequences:
2, 4, 8, 16, 32, 64, .......
243, 81, 27, 9, 3, 1, ...............
A geometric sequence is a sequence where each term is found by multiplying or dividing the same value from
one term to the next. We call this value "common ratio"
Looking at 2, 4, 8, 16, 32, 64, ......., carefully helps us to make the following observation:
As you can see, each term is found by multiplying 2, a common ratio to the previous term
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 term
Looking at 243, 81, 27, 9, 3, 1, ...............carefully helps us to make the following observation:
This time, to find each term, we divide by 3, a common ratio, from the previous term
Many geometric sequences can me modeled with an exponential function
an exponential function is a function of the form a^{n} where a
? 1
Here is the trick or recipe per se!
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 2^{1}, 2^{2}, 2^{3}, ...
We can therefore model the sequence with the following formula:
2^{n}
Check:
When n = 1, which represents the first term, we get 2^{1} = 2
When n = 2, which represents the second term, we get 2^{2} = 2 × 2 = 4
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 3^{5}, 3^{4}, 3^{3}, ...
We just have to model the sequence: 5, 4, 3, .....
The process will be briefly explained here. For detailed explanation, see arithmetic sequence
The number we subtract to each term is 1
The number that comes right before 5 in the sequence is 6
We can therefore model the sequence with the following formula:
-1× n + 6
We can therefore model 243, 81, 27, 9, 3, 1, ............... with the exponential function below:
3^{-n + 6}
Check:
When n = 1, which represents the first term, we get 3^{-1 + 6}= 3^{5} = 243
When n = 2, which represents the second term, we get 3^{-2 + 6}= 3^{4} = 81