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There are many situations where a quantity increases or decreases by a constant factor, like interest rates, population growth, etc. These situations can be modeled with a special type of sequence called a *geometric sequence*. This lesson will introduce geometric sequences for specific everyday life applications and will show how they can be described using *explicit rules* and *recursive rules*.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Explore

A regular sheet of paper has a thickness of about $0.1$ millimeter. Every time the paper is folded in half, its thickness doubles. Try the applet below to explore how the thickness increases as the sheet keeps getting folded in half.

Challenge

Using the same applet, now try to predict values according to the pattern.

- If a regular sheet of paper with a thickness of $0.1$ millimeters could be folded in half $15$ times, how tall would it be? Would it be taller than an average person? For reference, the height of an average person is $1.7$ meters.
- If it could be folded in half $20$ times, how tall would it be? Would it be taller than a $10$ story building? For reference, a $10$ story building is about $45$ meters tall.

Discussion

As can be seen in the previous applet, each time the sheet of paper was folded in half, its thickness doubled. The values for the paper thickness after each fold can be represented by terms of a specific type of sequence called a *geometric sequence*.

Concept

A geometric sequence is a sequence in which the ratio $r$ between consecutive terms is a nonzero constant. This ratio is called the common ratio. The following is an example geometric sequence with first term $3$ and common ratio $2.$

The behavior of geometric sequences generally depends on the values of the first term $a_{1}$ and the common ratio $r.$ The following table shows the effects of these parameters.

$a_{1}>0$ | $a_{1}<0$ | |
---|---|---|

$r>1$ | Increasing $3→×26→×212→×224→×248…$ |
Decreasing $-3→×2-6→×2-12→×2-24→×2-48…$ |

$r=1$ | Constant
$3→×13→×13→×13→×13…$ |
Constant
$-3→×1-3→×1-3→×1-3→×1-3…$ |

$0<r<1$ | Decreasing $48→×21 24→×21 12→×21 6→×21 3…$ |
Increasing $-48→×21 -24→×21 -12→×21 -6→×21 -3…$ |

$r<0$ | Alternating
$3→×(-2)-6→×(-2)12→×(-2)-24→×(-2)48…$ |
Alternating
$-3→×(-2)6→×(-2)-12→×(-2)24→×(-2)-48…$ |

Like for any other sequence, the first term of a geometric sequence is denoted by $a_{1},$ the second by $a_{2},$ and so on. Since geometric sequences have a common ratio $r,$ once one term is known, the next term can always be found by multiplying the known term by $r.$

In fact, the sequence can be found using only $a_{1}$ and $r,$ since all the subsequent terms can be found by multiplying $a_{1}$ by $r$ a specific number of times. Because of this, geometric sequences have the following general form.

$a_{1},a_{1}r,a_{1}r_{2},a_{1}r_{3},a_{1}r_{4},…$

Pop Quiz

Discussion

Geometric sequences can be described by using a formula that uses the positions of the terms to calculate their values. This formula is called an *explicit rule* of the geometric sequence.

Rule

Every geometric sequence can be described by a function known as the explicit rule, whose input is the position of a term $n$ and whose output is the term's value $a_{n}.$ An explicit rule for a geometric sequence has the following general form.

$a_{n}=a_{1}⋅r_{n−1}$

Here, $a_{1}$ is the first term of the sequence and $r$ is the common ratio.

Recall that every geometric sequence has a common ratio $r.$ Therefore, it is possible to find every term of the sequence by multiplying the first term $a_{1}$ by this common ratio a particular number of times. Therefore, knowing $a_{1}$ and $r$ is enough to describe the whole geometric sequence.

It is easier to identify a pattern that can be used to write a general expression for the explicit rule by making a table. Note that by the Zero Exponent Property, $r_{0}$ is equal to $1.$ Furthermore, $r$ can be written as $r_{1}.$

$n$ | $a_{n}$ | Using $a_{1}$ and $r$ |
---|---|---|

$1$ | $a_{1}$ | $a_{1}⋅r_{0}$ |

$2$ | $a_{2}$ | $a_{1}⋅r_{1}$ |

$3$ | $a_{3}$ | $a_{1}⋅r_{2}$ |

$4$ | $a_{4}$ | $a_{1}⋅r_{3}$ |

It can be seen that the exponent of the common ratio is always $1$ less than the value of the position $n.$ With this pattern, it is possible to write the explicit rule in the same form as the formula given at the beginning.

$a_{n}=a_{1}⋅r_{n−1}$

Example

Jordan is studying her biology notes. She finds out that a bacterium can divide into two bacteria in a period of time of about $20$ minutes. These two bacteria can then divide into two bacteria each, and so on.

Having read her notes, she is now ready for the lab practice. In a glass slide she has prepared a sample with $7$ isolated bacteria.

For her practice, Jordan had to check every $20$ minutes and count the number of bacteria. The results can be written as a sequence.

a Show that this situation can be modeled by using a geometric sequence.

b Find the next three terms of the sequence.

a **Demonstration:** See solution.

b **Terms:** $a_{5}=112,a_{6}=224,a_{7}=448$

a Recall that a sequence is geometric if it has a common ratio.

b In a geometric sequence, any term can be found by multiplying the preceding term by the common ratio.

a Recall that a sequence is geometric if it has a common ratio. Since the the number of bacteria is doubling after every $20$ minutes, the common ratio for this sequence is $r=2.$

Since this sequence has a common ratio, it is, by definition, a geometric sequence.

b Once one term is known, the next term can be found by multiplying the known term by the common ratio $r.$ For example, knowing that $a_{4}=56,$ the next term $a_{5}$ can be obtained by multiplying $56$ by the value of $r.$

$a_{5}=a_{4}r$

Substitute values and evaluate

$a_{5}=112$

Example

Ramsha's ecology teacher asks each of the $30$ students in her class to plant a seed. Then, they explain that if each student asked $3$ people to do the same by tomorrow, and these $3$ people did the same by the next day, the amount of planted seeds could modeled by a geometric sequence with the explicit rule $a_{n}=30⋅3_{n−1}.$

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b On the $10_{th}$ day, how many seeds would be planted?

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a The explicit rule must be evaluated for $n=5$ since the number of seeds planted on the $5_{th}$ day is to be calculated.

b The explicit rule must be evaluated for $n=10$ since the number of seeds planted on the $10_{th}$ day is to be calculated.

a To find the number of seeds planted on a specific day, the day number should be used as the $n-$value in the explicit rule. Therefore, to find the number of seeds planted on the $5_{th}$ day, the rule will be evaluated for $n=5.$

$a_{n}=30⋅3_{n−1}$

Substitute $5$ for $n$ and evaluate

Substitute

$n=5$

$a_{5}=30⋅3_{5−1}$

SubTerm

Subtract term

$a_{5}=30⋅3_{4}$

CalcPow

Calculate power

$a_{5}=30⋅81$

Multiply

Multiply

$a_{5}=2430$

b Just as in the previous part, the explicit rule will be evaluated for the $n-$value that represents the desired day — in this case, $n=10.$

$a_{n}=30⋅3_{n−1}$

Substitute $10$ for $n$ and evaluate

Substitute

$n=10$

$a_{10}=30⋅3_{10−1}$

SubTerm

Subtract term

$a_{10}=30⋅3_{9}$

CalcPow

Calculate power

$a_{10}=30⋅19683$

Multiply

Multiply

$a_{10}=590490$

Example

There is a famous story about the invention of chess. When the game was presented to the king, he was so happy about it that he told the inventor to choose any payment. The inventor asked the king to put a single grain of rice on the first square, two grains on the second, four on the third and so on. The amount on the final square was the desired payment.

The king was surprised and believed that this was such a bad decision for the inventor, as the king thought this debt could be paid with no more than a bag of rice. However, when he ordered his treasurer to pay the agreed amount, it turned out that this wealthy king was not rich enough as to pay the debt. In fact, it is impossible for anyone to pay it!

a To explore this in detail, first find an explicit rule to model this situation.

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b Find the how many of grains of rice are needed to pay this debt if a board of chess has $64$ squares. Give the answer, rounded to $3$ significant figures.

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c A single grain of rice weighs about $4.4×10_{-6}$ kilograms. The amount of rice produced in the entire world annually is about $480$ million metric tons, or $4.8×10_{11}$ kilograms. If the global annual rice production were used, how many years would be needed to pay this debt? Round the answer to the nearest year.

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a The general form for the explicit rule of a geometric sequence is $a_{n}=a_{1}⋅r_{n−1}.$

b Evaluate the rule found in Part A for $n=64.$

c First calculate the weight of the total amount of grains of the debt, then find the years needed to produce that amount.

a This situation can be modeled by a sequence with the first term $a_{1}=1,$ since a single grain is used on the first square, and a common ratio $r=2,$ since the terms double every time. First, the general form for the explicit rule of a geometric sequence will be recalled.

$a_{n}=a_{1}⋅r_{n−1} $

Now, the values $a_{1}=1$ and $r=2$ will be substituted to find the explicit rule of the presented sequence.
$a_{n}=a_{1}⋅r_{n−1}⇓a_{n}=1⋅2_{n−1} $

b To find the total number of grains of rice of the king's debt, the explicit formula will be evaluated for $n=64,$ since the chessboard has $64$ squares.

$a_{n}=1⋅2_{n−1}$

Substitute $64$ for $n$ and evaluate

Substitute

$n=64$

$a_{64}=1⋅2_{64−1}$

SubTerm

Subtract term

$a_{64}=1⋅2_{63}$

CalcPow

Calculate power

$a_{64}=1⋅9.223372…×10_{18}$

OneMult

$1⋅a=a$

$a_{64}=9.223372…×10_{18}$

RoundSigDig

Round to $3$ significant digit(s)

$a_{64}=9.22×10_{18}$

c First the total weight of the debt will be found. To do this, the number of grains will be multiplied by the weight of each individual grain: $4.4×10_{-6}$ kilograms.

$(9.22×10_{18})(4.4×10_{-6})$

Simplify

CommutativePropMult

Commutative Property of Multiplication

$(9.22)(4.4)(10_{18})(10_{-6})$

Multiply

Multiply

$40.568(10_{18})(10_{-6})$

MultPow

$a_{m}⋅a_{n}=a_{m+n}$

$40.568(10_{12})$

Write in scientific notation

$4.0568×10_{13}$

$4.8×10_{11}4.0568×10_{13} $

$85$

Example

A criminal mastermind started a big scam. They emailed some number of people and convinced them to send money by providing suspicious information for an investment plan. The criminal told each victim that all they needed to do was to contact $5$ people and ask them to send the same amount of money, and the mastermind's company would take care of the rest.

The scammer gave each victim one week to find $5$ people. Then, they gave one week to the new people to repeat the process. However, because of a blunder, the police caught the criminal.

a Which of the following options represents an explicit rule that can be used to find the number of new victims of this pyramid scheme at a given week?

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style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.964108em;vertical-align:-0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord\">5<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">n<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8388800000000001em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">5<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">2<\/span><span class=\"mord\">5<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"minner\">\u2026<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8388800000000001em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\">5<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\">5<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">2<\/span><span class=\"mord\">5<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mspace\" style=\"margin-right:1em;\"><\/span><span class=\"minner\">\u2026<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.58056em;vertical-align:-0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"><\/span><span class=\"mord\">5<\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">r<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathdefault mtight\">n<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

b The criminal refuses to talk, but the police have gathered enough evidence to conclude that the operation has been going on for $4$ weeks, and that on the fourth week, the criminal received $$200000$ from the victims. Moreover, from some victim statements, the police also know that each victim was asked to send $$80.$ What was the starting number of victims?

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a The general form for the explicit rule of a geometric sequence is $a_{n}=a_{1}⋅r_{n−1}.$

b Since the total amount of money received together with the amount that each victim sent is known, the number of victims at the fourth week can be found. Use this information and the result from Part A to calculate the initial number of victims.

a The formula of an explicit rule for a geometric sequence with common ratio $r$ and initial term $a_{1}$ has the following form.

$a_{n}=a_{1}⋅r_{n−1} $

Here, $n$ is the position of the $n_{th}$ term and $a_{n}$ is the $n_{th}$ term's value. Since each victim had to contact $5$ people, who would then contact $5$ more people, and so on, it can be concluded that each week the number of victims increased $5$ times. Thus, the common ratio is $r=5.$ This value will be substituted into the above formula.
$a_{n}=a_{1}⋅r_{n−1}⇓a_{n}=a_{1}⋅5_{n−1} $

In this explicit rule, $a_{n}$ represents the total number of victims on the $n_{th}$ week. So far, the initial number of victims $a_{1}$ is unknown. b It is known that on the $4_{th}$ week, the scammer received a total amount of $$200000$ from the victims. Because $a_{4}$ represents the number of victims on the $4_{th}$ week, and since each victim sent $$80,$ $80a_{4}$ should equal the total amount received.

$80a_{4}=200000⇒a_{4}=2500 $

Now that $a_{4}$ is known, the value $n=4$ will be used in the explicit rule from Part A to determine the starting number of victims $a_{1}.$ $a_{n}=a_{1}⋅5_{n−1}$

Substitute

$n=4$

$a_{4}=a_{1}⋅5_{4−1}$

Substitute

$a_{4}=2500$

$2500=a_{1}⋅5_{4−1}$

Solve for $a_{1}$

SubTerm

Subtract term

$2500=a_{1}⋅5_{3}$

CalcPow

Calculate power

$2500=a_{1}⋅125$

DivEqn

$LHS/125=RHS/125$

$20=a_{1}$

RearrangeEqn

Rearrange equation

$a_{1}=20$

Discussion

It has been shown how an explicit rule can describe a geometric sequence with a function that receives the term position as input and returns the term's value as output. However, a geometric sequence can also be described with a *recursive relation*.

Concept

A recursive rule of a geometric sequence is a pair made of a recursive equation telling how the term $a_{n}$ is related to its preceding term $a_{n−1},$ and the first term of the sequence $a_{1}.$

$a_{1},a_{n}=a_{n−1}⋅r$

Note that if the first term is not given, the recursive equation by itself describes all different geometric sequences with the same common ratio.
This is why the first term should be specified in the recursive rule to uniquely define the specific geometric sequence.

Now it will be explained, step by step, how to write the recursive rule for a geometric sequence.

Method

The recursive rule of a geometric sequence includes the first term of the sequence and a recursive equation.

$a_{1},a_{n}=a_{n−1}⋅r$

$a$