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In the arithmetic sequences section, we introduced the idea of sequences which grow in a linear way, increasing by the same numerical value from term to term. Yet just as linear equations are far from the only kind of function, arithmetic sequences are far from the only kind of sequence!
The diagram to the right depicts a bouncing ball. How high is the ball's
first bounce?
second bounce?
fourth bounce?
How high do you think its tenth bounce will be?
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Discussion
Let's think about some sequences that follow a similar kind of pattern to the one above. Consider these:
What do the three sequences above have in common?
Each sequence is produced by multiplying by the same fixed number to get from one term to the next.
Sequence 1:
Sequence 2:
Sequence 3:
In each case, you “step” through the sequence by multiplying by the same number every time.
These are all examples of geometric sequences. Generally, a sequence is said to be geometric if we multiply by a constant factor to go from any one term to the next.
The number we multiply by is called the common ratio, and is denoted r.
Exercise
Consider the following sequences:
State the common ratio r of the one that is geometric.
Select the correct option
Identifying Geometric Sequences
A sequence is geometric if the ratio between consecutive terms is always constant, i.e.
We call r the common ratio.
For instance, the sequence
is geometric with r=41, but
is not since 69=23=2=36.
Exercise
For what value(s) of k is the sequence 3,k+1,12 geometric?
Select the correct option
Discussion
Imagine a geometric sequence with first term u1=81 and common ratio 2.
Can you find the second and third terms?
How would you find the twelfth term?
Each term is twice the one before. Starting from
we get
To reach the twelfth term, we keep doubling twelve times. In other words
and since
the twentieth term is 512.
A geometric sequence starts with a first term u1 and each consecutive term is multiplied by r:
Notice that since each term un is defined in terms of the last term un−1, we are essentially multiplying u1 by r to the power of (n−1) every time.
General Term of a Geometric Sequence
The nth term of a geometric sequence is given by
where u1 is the first term and r is the common ratio.
Exercise
A geometric sequence has third term v3=2 and sixth term v6=−54.
Find the 11th term, v11.
Select the correct option
Example
A geometric sequence has u2=9 and u5=31. Find the first term and the common ratio.
Using the formula, we have
Dividing u5 by u2:
so
Now since u2=u1r=9, u1=r9=9⋅3=27.
Discussion
Geometric sequences appear all the time in real life, but they're usually not talked about in terms of a "first term" and "common ratio." In fact, one common way we talk about geometric sequences is with percentage increase or decrease.
Think about a small town whose population was 500 in the year 2000, and whose population has been steadily increasing ever since, at a rate of 2% per year.
How many people lived in the town in 2001? 2002? How about in 2015?
Try to talk about the population's growth using the language of geometric sequences. What is the first term u1? What is the common ratio r? Can you give an equation for the general term un?
The population in 2000 is the first term of a geometric sequence. A steady 2 % annual increase means each year’s population is 1.02 times the previous year’s.
Hence the general term is
where n=1 corresponds to 2000, n=2 to 2001, and so on.
For 2001 (n=2):
For 2002 (n=3):
For 2015 (n=16):
Repeated percent change is a form of geometric sequence. In the population example, the percent increase of 2% is applied to the previous year's population, not the original population.
In general, if you start with an amount A and increase each term by p%, then the common ratio is
If you start with an amount A and decrease each term by p%, then the common ratio is
This gives exactly a geometric sequence,
!
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