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Geometric Sequences

Writing Recursive Rules for Geometric Sequences

When a sequence is defined by a recursive rule, each term can be found using one or more preceding terms.

Method

Writing a Recursive Rule for a Geometric Sequence

The recursive rule for a geometric sequence can be expressed as
where r is the common ratio of the sequence. To write the rule, r must be found. Consider the geometric sequence
To find r, we can divide any term with the term that comes before it. Let's use a2=6 and a1=3.
Using r=2, the recursive rule can be written as
As it's written, this rule describes any sequence with a common ratio of 2. For example, it can describe as well as To ensure the recursive rule defines the given sequence, it is necessary to also give the first term, a1. Thus, the recursive rule for the given sequence is
Now that the recursive rule is known, it can be used to find any term, provided that the previous term is given. For example, since a4=24, a5 can be found.
Thus, the 5th term in the sequence is 48.
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Exercise
A team of researchers have studied a group of glaciers in the Himalayas over a four year period. Each year they calculated the approximate volume of the glaciers. Their results are shown.
Create a recursive rule that describes the volume of the glaciers. Then use the rule to find the volume of the glaciers in year 6, provided that they continue shrinking by the same ratio.
Show Solution
Solution
Before writing the recursive rule, we must make sure that the sequence is geometric. Let's determine the common ratio between two consecutive terms. For the first two terms it's given by
The common ratio between the successive terms is calculated in the same way.
We can see that each year the glaciers shrink by a factor 0.9. Therefore, the volumes of the glaciers form a geometric sequence with a common ratio 0.9. Using r and a1 the recursive rule can be written.
We can use this rule to determine the volume in year 6. We know a4=2187 km3. Since we're using a recursive rule, we must first find a5, then we can find a6.
If the glaciers continue shrinking by the same ratio, their volume in year 6 will approximately be
1771 km3.

Method

Translating between Explicit and Recursive Rules

As has been explored, it is possible to express sequences with explicit rules and recursive rules. Since the rules present different information about the sequence, it can be useful to translate between the two.
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Exercise
A geometric sequence is given by the explicit rule
Translate this to a recursive rule.
Show Solution
Solution
A recursive rule for a geometric sequence is written as
where a1 is the first term and r is the common ratio. Since an explicit rule for a geometric sequence is written as it can be seen that a1=42 and r=1.5. Thus we can write the recursive rule for this sequence as sequence as
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Exercise
A geometric sequence is defined by a recursive rule specified as
Write an explicit rule that describes the same sequence.
Show Solution
Solution
The explicit rule of a geometric sequence is written as
where a1 is the first term and r is the common ratio. To write an explicit rule, both a1 and r are needed. From the recursive rule
it can be seen that a1=3 and r=2. Thus the explicit rule can be written as
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