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

The recursive rule for a geometric sequence can be expressed as $a_{n}=r⋅a_{n−1},$ where $r$ is the common ratio of the sequence. To write the rule, $r$ must be found. Consider the geometric sequence $3,6,12,24,…$ To find $r,$ we can divide any term with the term that comes before it. Let's use $a_{2}=6$ and $a_{1}=3.$ $r=a_{1}a_{2} =36 =2$ Using $r=2,$ the recursive rule can be written as $a_{n}=2⋅a_{n−1}.$ As it's written, this rule describes any sequence with a common ratio of $2.$ For example, it can describe $-0.25,-0.5,-1,-2,…$ as well as $10,20,40,80,160,…$ To ensure the recursive rule defines the given sequence, it is necessary to also give the first term, $a_{1}.$ Thus, the recursive rule for the given sequence is $ a_{1}=3a_{n}=2⋅a_{n−1}. $ 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 $a_{4}=24,$ $a_{5}$ can be found. $a_{n}a_{5}a_{5}a_{5} =2⋅a_{n−1}=2⋅a_{4}=2⋅24=48 $

Thus, the $5th$ term in the sequence is $48.$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. $year1year2year3year4 :3000km_{3}:2700km_{3}:2430km_{3}:2187km_{3} $ 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

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 $a_{1}a_{2} =30002700 =0.9.$ 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 $a_{1}$ the recursive rule can be written. $a_{1}=3000a_{n}=0.9⋅a_{n−1} $ We can use this rule to determine the volume in year $6.$ We know $a_{4}=2187km_{3}.$ Since we're using a recursive rule, we must first find $a_{5},$ then we can find $a_{6}.$ $ a_{5}=0.9⋅a_{4}=0.9⋅2187=1968.3km_{3}a_{6}=0.9⋅a_{5}=0.9⋅1968.3=1771.47km_{3} $ If the glaciers continue shrinking by the same ratio, their volume in year $6$ will approximately be $1771km_{3}.$

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.

A geometric sequence is given by the explicit rule $a_{n}=42⋅1.5_{n−1}.$ Translate this to a recursive rule.

Show Solution

A recursive rule for a geometric sequence is written as $ a_{1}=...a_{n}=r⋅a_{n−1} $ where $a_{1}$ is the first term and $r$ is the common ratio. Since an explicit rule for a geometric sequence is written as $a_{n}=a_{1}⋅r_{n−1},$ it can be seen that $a_{1}=42$ and $r=1.5.$ Thus we can write the recursive rule for this sequence as sequence as $a_{1}=42a_{n}=1.5⋅a_{n−1}. $

A geometric sequence is defined by a recursive rule specified as $a_{1}=3a_{n}=2⋅a_{n−1}. $ Write an explicit rule that describes the same sequence.

Show Solution

The explicit rule of a geometric sequence is written as $a_{n}=r⋅a_{n−1},$ where $a_{1}$ is the first term and $r$ is the common ratio. To write an explicit rule, both $a_{1}$ and $r$ are needed. From the recursive rule $a_{1}=3a_{n}=2⋅a_{n−1} $ it can be seen that $a_{1}=3$ and $r=2.$ Thus the explicit rule can be written as $a_{n}=3⋅2_{n−1}.$

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