8. Geometric Sequences
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Chapter 6
8.
Geometric Sequences
Geometric sequences are sequences where the ratio between consecutive terms remains constant. This unique characteristic allows them to model various phenomena in the real world, such as interest rates, population growth, and more. The essence of a geometric sequence is its common ratio, which dictates how each term relates to the previous one. For instance, if a bacterium divides every 20 minutes, the number of bacteria can be represented as a geometric sequence with a common ratio of 2. Similarly, when a sheet of paper is folded, its thickness doubles, making it another example of a geometric sequence. By understanding these sequences, one can predict outcomes, model growth or decay, and gain insights into various mathematical and real-world scenarios. They serve as a powerful tool in mathematics, offering both theoretical knowledge and practical applications.
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| | 15 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Geometric Sequences
Slide of 15
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
How Does the Thickness of a Sheet of Paper Increase as It Is Folded in Half?
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
Predicting the Height of a Paper Sheet Folded Multiple Times
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
Introducing Geometric Sequences
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
Geometric Sequence
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 a1 and the common ratio r. The following table shows the effects of these parameters.
| a1>0 | a1<0 | |
|---|---|---|
| r>1 | Increasing 3 →×2 6 →×2 12 →×2 24 →×2 48… |
Decreasing -3 →×2 -6 →×2 -12 →×2 -24 →×2 -48… |
| r=1 | Constant
3 →×1 3 →×1 3 →×1 3 →×1 3… |
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 a1, the second by a2, 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 a1 and r, since all the subsequent terms can be found by multiplying a1 by r a specific number of times. Because of this, geometric sequences have the following general form.
a1, a1r, a1r2, a1r3, a1r4, …
Pop Quiz
Identifying Geometric Sequences
Discussion
Describing Geometric Sequences Using an Explicit Rule
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
Explicit Rule of Geometric Sequences
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 an. An explicit rule for a geometric sequence has the following general form.
an=a1⋅rn−1
Here, a1 is the first term of the sequence and r is the common ratio.
Proof
Proof by InductionRecall 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 a1 by this common ratio a particular number of times. Therefore, knowing a1 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, r0 is equal to 1. Furthermore, r can be written as r1.
| n | an | Using a1 and r |
|---|---|---|
| 1 | a1 | a1⋅r0 |
| 2 | a2 | a1⋅r1 |
| 3 | a3 | a1⋅r2 |
| 4 | a4 | a1⋅r3 |
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.
an=a1⋅rn−1
Example
Modeling Multiplication of Bacteria Using Geometric Sequences
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.
Answer
a Demonstration: See solution.
b Terms: a5=112,a6=224,a7=448
Hint
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.
Solution
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 a4=56, the next term a5 can be obtained by multiplying 56 by the value of r.
a5=a4r
▼
Substitute values and evaluate
a5=112
Example
An Ecology Project that Follows a Geometric Sequence
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 an=30⋅3n−1.
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b On the 10th day, how many seeds would be planted?
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Hint
a The explicit rule must be evaluated for n=5 since the number of seeds planted on the 5th day is to be calculated.
b The explicit rule must be evaluated for n=10 since the number of seeds planted on the 10th day is to be calculated.
Solution
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 5th day, the rule will be evaluated for n=5.
an=30⋅3n−1
▼
Substitute 5 for n and evaluate
Substitute
n=5
a5=30⋅35−1
SubTerm
Subtract term
a5=30⋅34
CalcPow
Calculate power
a5=30⋅81
Multiply
Multiply
a5=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.
an=30⋅3n−1
▼
Substitute 10 for n and evaluate
Substitute
n=10
a10=30⋅310−1
SubTerm
Subtract term
a10=30⋅39
CalcPow
Calculate power
a10=30⋅19683
Multiply
Multiply
a10=590490
Example
The Geometric Sequence Behind the Story of Chess
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 4.8×1011 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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Hint
a The general form for the explicit rule of a geometric sequence is an=a1⋅rn−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.
Solution
a This situation can be modeled by a sequence with the first term a1=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.
an=a1⋅rn−1
Now, the values a1=1 and r=2 will be substituted to find the explicit rule of the presented sequence.
an=a1⋅rn−1⇓an=1⋅2n−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.
an=1⋅2n−1
▼
Substitute 64 for n and evaluate
Substitute
n=64
a64=1⋅264−1
SubTerm
Subtract term
a64=1⋅263
CalcPow
Calculate power
a64=1⋅9.223372…×1018
OneMult
1⋅a=a
a64=9.223372…×1018
RoundSigDig
Round to 3 significant digit(s)
a64=9.22×1018
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.
(9.22×1018)(4.4×10-6)
▼
Simplify
CommutativePropMult
Commutative Property of Multiplication
(9.22)(4.4)(1018)(10-6)
Multiply
Multiply
40.568(1018)(10-6)
MultPow
am⋅an=am+n
40.568(1012)
Write in scientific notation
4.0568×1013
4.8×10114.0568×1013
85
Example
Investigating a Police Case Using Geometric Sequences
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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Hint
a The general form for the explicit rule of a geometric sequence is an=a1⋅rn−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.
Solution
a The formula of an explicit rule for a geometric sequence with common ratio r and initial term a1 has the following form.
an=a1⋅rn−1
Here, n is the position of the nth term and an is the nth 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.
an=a1⋅rn−1⇓an=a1⋅5n−1
In this explicit rule, an represents the total number of victims on the nth week. So far, the initial number of victims a1 is unknown. b It is known that on the 4th week, the scammer received a total amount of $200000 from the victims. Because a4 represents the number of victims on the 4th week, and since each victim sent $80, 80a4 should equal the total amount received.
80a4=200000⇒a4=2500
Now that a4 is known, the value n=4 will be used in the explicit rule from Part A to determine the starting number of victims a1. an=a1⋅5n−1
Substitute
n=4
a4=a1⋅54−1
Substitute
a4=2500
2500=a1⋅54−1
▼
Solve for a1
SubTerm
Subtract term
2500=a1⋅53
CalcPow
Calculate power
2500=a1⋅125
DivEqn
LHS/125=RHS/125
20=a1
RearrangeEqn
Rearrange equation
a1=20
Discussion
Defining Geometric Sequences Recursively
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
Recursive Rule of a Geometric Sequence
A recursive rule of a geometric sequence is a pair made of a recursive equation telling how the term an is related to its preceding term an−1, and the first term of the sequence a1.
a1,an=an−1⋅r
Now it will be explained, step by step, how to write the recursive rule for a geometric sequence.
Method
Writing a Recursive Rule for a Geometric Sequence
The recursive rule of a geometric sequence includes the first term of the sequence and a recursive equation.
a1,an=an−1⋅r
a12,a26,a318,a454,…
There are 3 steps to follow to write the recursive rule for this geometric sequence.
1
Find the Common Ratio
expand_more To find the common ratio r, identify any two consecutive terms and divide the latter term by the former one.
Since each term increases by factor of r, a2 is equal to a1r and a4 is equal to a1r3.
Note that when solving for r, only the principal root was considered because it was known from the beginning that r is positive.
a12,a26,a318,a454,…618=3 ⇒ r=3(lal
What to Do When No Consecutive Terms Are Known
The common ratio can still be found even if the sequence is known to be geometric but no consecutive terms are known. In general, it is enough to know any two terms of a geometric sequence and their positions. For example, suppose that it is known that r is positive and only a2 and a4 are known.a1?,a26,a3?,a454,…
Recall the general form of a geometric sequence. a4=a1r3 a2=a1r3 ⇒ 54=a1r3⇒ 46=a1r
A single equation in terms of only r can be written by dividing the corresponding sides of the resulting equations. This equation can then be solved.
654=a1ra1r3
▼
Simplify
CalcQuot
Calculate quotient
9=a1ra1r3
CancelCommonFac
Cancel out common factors
9=a1 ra1 r3
SimpQuot
Simplify quotient
9=rr3
aam=am−1
9=r2
▼
Solve for r
r=3
2
Identify the First Term of the Sequence
expand_more To completely define the geometric sequence, the first term should also be determined. Without it, the recursive equation would describe any geometric sequence with common ratio r=3. From the sequence, it can be seen that a1=2.
a12,a26,a318,a454,…
What to Do If a1 Is Not Known
Reconsider the case where only a2 and a4 are known. Two equations were obtained.5446=a1r3=a1r
Since it was determined that r=3, this value can be substituted into either equation to find a1. This will be illustrated using the latter equation.
6=a13⇒a1=2
3
Write the Equation Rule
expand_more Finally, the recursive rule for the sequence can be written by putting together the previous results and substituting the common ratio and the value of the first term into the general formula.
a1,an=an−1⋅r⇓a1=2,an=an−1⋅3
Example
Using a Recursive Rule for Carbon Dating a Sample
The study of the elements present in the atmosphere makes carbon dating possible. In particular, the ratio R of carbon-14 to carbon-12 is of main importance.
As can be seen, the position of the corresponding term is n=4. This means that in order for the sample to have only 6.25% of the original ratio R left, 4 half life periods must have passed. The age of the sample will be found by multiplying 4 and the length of the half life period, which is given to be 5730 years.
carbon-12carbon-14=R
Since carbon-14 is radioactive, R reduces by half after a specific period of time called a half life. Because organisms take carbon from the atmosphere while breathing, the same carbon ratio of the atmosphere is present in their bodies while they are alive. However, when they die, they stop inhaling carbon from the atmosphere and the ratio starts to decrease as the carbon-14 decays over time. The diagram above shows a sequence representing the percentage of the original ratio of carbon R present in the organism. The term's value an is the percentage left of the original ratio R in the sample after n half life periods.
a Prove that this sequence is geometric and find a recursive rule for it.
b Use the recursive rule to find the next two terms of the sequence.
c An examined sample has only 6.25% of the original carbon ratio from the atmosphere. Use the terms found in the Part B to estimate the age of the sample, considering that the half life period of carbon-14 is about 5730 years.
Answer
a Proof: See solution.
Recursive Rule: a1=50,an=an−1⋅21
Recursive Rule: a1=50,an=an−1⋅21
b Terms: a4=6.25,a5=3.125
c Estimated Age: 22920 years
Hint
a The general form for the recursive rule of a geometric sequence is a1, an=an−1⋅r
b Use the recursive rule found in Part A. To find a specific term, the previous term needs to be known.
c Note that the positions of the terms represent the number of half life periods that have passed.
Solution
a It is known that the percentage decreases by half from one term to the next. Therefore, the sequence has a common ratio of r=21. Because it has a common ratio, it is, by definition, a geometric sequence. Recall the form of a general recursive rule for a geometric sequence.
a1,an=an−1⋅r
Since the common ratio is already known, the first term a1 must be identified. This can be found in the sequence. Now the values of the first term and the common ratio will be substituted in the general form to obtain the desired recursive rule.
a1,an=an−1⋅r⇓a1=50,an=an−1⋅21
b The next two terms can be found by using the recursive formula consecutively. First a4 will be found by substituting n=4 and a3=12.5.
an=an−1⋅21
▼
Substitute values and evaluate
Substitute
n=4
a4=a4−1⋅21
SubTerm
Subtract term
a4=a3⋅21
Substitute
a3=12.5
a4=12.5⋅21
MoveLeftFacToNumOne
a⋅b1=ba
a4=212.5
CalcQuot
Calculate quotient
a4=6.25
| a1=50,(an=an−1⋅21 | ||
|---|---|---|
| n | Substitution | an |
| n=4 | a4=a3⋅21⇓a4=12.5⋅21
|
a4=6.25 |
| n=5 | a5=a4⋅21⇓a5=6.25⋅21
|
a5=3.125 |
Now, the next two terms can be added to the sequence.
c The first thing to do is to identify the position of the term 6.25 in the sequence.
Age of the Sample4×5730=22920
The estimated age of the sample is 22920 years.
Example
Modeling a Bouncing Ball Using Geometric Sequences
Ali is curious about what would happen if he dropped a ball from the top of the building he lives in, which is about 81 meters tall. He has noticed that when the ball bounces, it reaches one third the height it was dropped from, so the ball would reach first 381=27 meters high on the first bounce. To calculate the height for the next bounces, he writes a recursive rule.
a Use the recursive rule to calculate the values of the first three terms.
b Find the corresponding explicit rule and calculate the height of the 5th bounce. Give the answer rounded to two decimal places.
c Ali did some research online and found a bouncing ball of a special material that, when dropped from the same height of 81 meters, its bouncing height can be modeled according to the following explicit rule.
an=72.9⋅0.9n−1
Find the corresponding recursive rule for this new bouncing ball.
Answer
a Terms: a1=27,a2=9,a3=3
b Explicit Rule: an=27⋅(31)n−1
Height of the Fifth Bounce: 0.33 meters.
Height of the Fifth Bounce: 0.33 meters.
c Recursive Rule: a1=72.9,an=an−1⋅0.9
Hint
a Use the given recursive rule repeatedly. Note that to find a specific term, the previous term should be known.
b Recall that the general form for the explicit rule of a geometric sequence is an=a1⋅rn−1.
c Use the formula for the recursive rule of a geometric sequence.
Solution
a The first three terms can be found by using the recursive rule repeatedly. It is already known that a1=27. Therefore, the next term a2 will be found by substituting n=2 and a1=27.
an=an−1⋅31
▼
Substitute values and evaluate
Substitute
n=2
a2=a2−1⋅31
SubTerm
Subtract term
a2=a1⋅31
Substitute
a1=27
a2=27⋅31
MoveLeftFacToNumOne
a⋅b1=ba
a2=327
CalcQuot
Calculate quotient
a2=9
| an=an−1⋅31( | ||
|---|---|---|
| n | Substitution | an |
| n=1 | a1=27
|
a1=27 |
| n=2 | a2=a1⋅31⇓a2=27⋅31
|
a2=9 |
| n=3 | a3=a2⋅31⇓a3=9⋅31
|
a3=3 |
b To find the explicit rule of the sequence, first identify the first term and the common ratio from the known recursive rule. In order to do this, start by reviewing the general form of a recursive rule.
a1,an=an−1⋅r
Here, a1 represents the first term of the sequence and r is the common ratio. Their values will be identified by comparing this general form and the given recursive rule.
General Recursive Rulea1,an=an−1⋅rParticular Recursive Rulea1=27,an=an−1⋅31
It can be concluded that a1=27 and r=31. Now the general form of an explicit rule will be shown and these values will be substituted to find the desired explicit rule for this situation.
General Explicit Rulean=a1⋅rn−1Particular Explicit Rulean=27⋅(31)n−1
Now, to find the fifth term a5, the explicit rule will be evaluated at n=5.
an=27⋅(31)n−1
▼
Substitute 5 for n and evaluate
Substitute
n=5
a5=27⋅(31)5−1
SubTerm
Subtract term
a5=27⋅(31)4
CalcPow
Calculate power
a5=27⋅811
MoveLeftFacToNumOne
a⋅b1=ba
a5=8127
CalcQuot
Calculate quotient
a5=0.333333…
RoundDec
Round to 2 decimal place(s)
a5=0.33
c Similar to the previous part, in order to write a recursive rule, the first term and the common ratio should be identified. To do this, the general form of an explicit rule will be reviewed.
an=a1⋅rn−1
Here, a1 represents the first term of the sequence and r is the common ratio. Now, the general equation will be compared to the specific explicit rule provided.
General Explicit Rulean=a1⋅rn−1Particular Explicit Rulean=72.9⋅0.9n−1
It can be concluded that a1=72.9 and r=0.9. Finally, the general form of a recursive rule will be written and these values will be substituted to find the recursive rule corresponding to the given explicit rule.
General Explicit Rulea1,an=an−1⋅rParticular Explicit Rulea1=72.9,an=an−1⋅0.9
Discussion
Geometric Sequences and Exponential Functions
Because the values of the terms of a geometric sequence increase by a constant factor, every geometric sequence shows an exponential relation, where the common ratio can be considered as the constant multiplier of the associated exponential function. Consider the following geometric sequence.
The sequence can be represented by a table of values, where the independent variable n represents the term's position and the dependent variable an represents the term's value.
In this case, the common ratio is 2. Therefore, all the terms of the sequence lie on the graph of an exponential curve with a constant multiplier of 2.
Closure
Modeling the Thickness of a Sheet of Paper using Geometric sequences
Now that geometric sequences have been introduced and it has been shown how to describe them by using explicit rules, the challenge at the beginning of the lesson will be solved. 
a If a regular sheet of paper, with a thickness of 0.1 millimeter, could be folded in half 15 times, how tall would it be? Write the answer in meters rounded to one decimal place. Would it be taller than an average person? For reference, the height of an average person is 1.7 meters.
b If it could be folded in half 20 times, how tall would it be? Write the answer in meters rounded to one decimal place. Would it be taller than a 10 story building? For reference, a 10 story building is about 45 meters tall.
Answer
a Height: 3.3 meters
Taller Than a Person? Yes
Taller Than a Person? Yes
b Height: 104.9 meters
Taller Than a 10 Story Building? Yes
Taller Than a 10 Story Building? Yes
Solution
a If the height of the paper is represented by the term value and the term's position represents the number of folds, the situation can be written as a sequence. Moreover, since the terms of this sequence double their value each time, this is a geometric sequence with a common ratio r=2.
From the sequence, it can be seen that the first term is a1=0.2. Since both the common ratio and the initial term are already known, they can be substituted into the formula of the explicit rule of a geometric sequence.
The folded sheet of paper would be 3276.8 millimeters tall. Since a meter has 1000 millimeters, the height in meters will be obtained by dividing it by 1000.
an=a1⋅rn−1⇓an=0.2⋅2n−1
Now, to find how tall the folded paper sheet would be after being folded 15 times, the explicit rule will be evaluated for n=15.
an=0.2⋅2n−1
▼
Substitute 15 for n and evaluate
Substitute
n=15
a15=0.2⋅215−1
SubTerm
Subtract term
a15=0.2⋅214
CalcPow
Calculate power
a15=0.2⋅16384
Multiply
Multiply
a15=3276.8
10003276.8=3.2768
This rounds to about 3.3 meters. Therefore, the folded paper would be definitely taller than an average person. b To find the how tall the folded paper sheet would be after being folded 20 times, the explicit rule will now be evaluate for n=20.
The folded sheet of paper would be 104857.6 millimeters tall. Now this quantity will be divided by 1000 to convert it to meters.
an=0.2⋅2n−1
▼
Substitute 20 for n and evaluate
Substitute
n=20
a15=0.2⋅220−1
SubTerm
Subtract term
a15=0.2⋅219
CalcPow
Calculate power
a15=0.2⋅524288
Multiply
Multiply
a15=104857.6
1000104857.6=104.8576
This rounds to about 104.9 meters. Therefore, it would be definitely taller than a 10 story building.
Geometric Sequences
Find the common ratio r of the geometric sequences.
4,16,64,256, …
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36,6,1,61, …
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0.1,1,10,100, …
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We want to find the common ratio of the given geometric sequence. The common ratio is the ratio of any two consecutive terms. Therefore, to find the common ratio, we must calculate the quotient between two consecutive terms.
As we can see, the common ratio for this sequence is 4.
Again, we want to find the ratio between consecutive terms. We can do so by calculating the quotient between two consecutive terms in the sequence.
The common ratio for this sequence is 16.
Similar to the previous parts, we calculate the common ratio by calculating the quotient between consecutive terms.
The common ratio for this sequence is 10.
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Determine whether each sequence is a geometric sequence.
-8,0,8,16, …
{"type":"choice","form":{"alts":["Geometric","Non-geometric"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
-1,4,-7,10, …
{"type":"choice","form":{"alts":["Geometric","Non-geometric"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
493,73,3,21, …
{"type":"choice","form":{"alts":["Geometric","Non-geometric"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
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We are want to determine whether the given sequence is geometric or not. In order to do so, we will calculate the ratio between consecutive terms.
As we can see, the ratios are not the same. One of them is even a division by zero, which is undefined! Therefore, the sequence is not geometric.
We will investigate whether the sequence is geometric by examining the ratios of consecutive terms as we did in Part A.
Again, the ratios are not the same. Therefore, the sequence is not geometric.
Once more, we will investigate the ratios of consecutive terms.
As we can see, the ratios between all pairs of consecutive terms is 7. Therefore, there is a common ratio and the sequence is geometric.
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Consider the recursive rule of a geometric sequence.
a1=2, an=3an−1
Find the first five terms of the sequence.
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The first term of the geometric sequence is already given as part of the recursive rule. a_1 = 2, a_n=3a_(n-1) Now that we know that a_1= 2, we can use the provided rule to find the second term in the sequence a_2. To do so, we will substitute 2 for n and simplify.
We can follow the same procedure to find the next three terms in the sequence, a_3, a_4, and a_5.
| n | a_n=3a_(n-1) | a_n |
|---|---|---|
| 1 | a_1=2 | 2 |
| 2 | a_2=3a_(2-1) ⇕ a_2=3 a_1 |
a_2=3( 2) ⇕ a_2= 6 |
| 3 | a_3=3a_(3-1) ⇕ a_3=3 a_2 |
a_3=3( 6) ⇕ a_3= 18 |
| 4 | a_4=3a_(4-1) ⇕ a_4=3 a_3 |
a_4=3( 18) ⇕ a_4= 54 |
| 5 | a_5=3a_(5-1) ⇕ a_5=3 a_4 |
a_5=3( 54) ⇕ a_5= 162 |
Therefore, the first five terms of the sequence are 2, 6, 18, 54, and 162. We could also find the consecutive terms by multiplying the previous term by 3.
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Write the next three terms of each geometric sequence.
5,25,125,625, …
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32,8,2,21, …
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916,38,4,6, …
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Let's start by determine the common ratio of the geometric sequence.
The common ratio is 5. To find the next three terms, we will use the fact that each term after a_1 is obtained by multiplying the previous term by the common ratio. lll a_5 = 625 ( 5) & ⇒ & a_5= 3125 a_6 = 3125 ( 5) & ⇒ & a_6=15 625 a_7 = 15 625 ( 5) & ⇒ & a_7=78 125 The next three terms are 3125, 15 625, and 78 125.
As we did before, we will start by finding the common ratio by calculating the quotient between consecutive terms.
The common ratio is 14. To find the next three terms, we multiply the previous term by 14. lll a_5 = 1/2 ( 1/4) & ⇔ & a_5=1/8 [0.5em] a_6 = 1/8 ( 1/4) & ⇔ & a_6=1/32 [0.5em] a_7 = 1/32 ( 1/4) & ⇔ & a_7=1/128 The next three terms are 18, 132, and 1128.
As in the previous parts, we start by calculating the common ratio.
The common ratio is 32. To find the next three terms, we multiply the previous term by 32. lll a_5 = 6 ( 3/2) & ⇔ & a_5=9 [0.5em] a_6 = 9 ( 3/2) & ⇔ & a_6=27/2 [0.5em] a_7 = 27/2( 3/2) & ⇔ & a_7=81/4 The next three terms are 9, 272, and 814.
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Find a9 by first writing an equation for the nth term of each geometric sequence.
2,8,32,128, …
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0.1,0.3,0.9,2.7, …
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15,3,53,253, …
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The explicit rule of a geometric sequence follows a specific format. a_n= a_1 r^(n-1) In this form, a_1 is the first term of the sequence, r its common ratio, and a_n the nth term in the sequence. From the sequence, we see that the first term is a_1= 2. Let's now determine the common ratio.
By substituting r= 4 and a_1= 2 into the formula, we can find the formula for this sequence. a_n= 2( 4)^(n-1) Finally, we substitute n= 9 into the formula and simplify to determine the value of the ninth term.
The 9th term in the sequence is 131 072.
As in Part A, we can immediately see that the first term is 0.1. Let's proceed by finding the common ratio between consecutive terms.
The common ratio is 3. Now we can write the equation by substituting r= 3 and a_1= 0.1 into the formula for a geometric sequence. a_n= 0.1( 3)^(n-1) Next, we substitute n= 9 into the equation and evaluate.
The ninth term is 656.1.
As in previous parts, we can identify the first term by looking at the sequence. This time the first term is 15. Let's now find the common ratio between consecutive terms.
The common ratio is 15. Now we have all the information we need to write the equation of the geometric sequence. \begin{gathered} a_{n}={\color{#0000FF}{15}}\left({\color{#FF0000}{\dfrac15}}\right)^{n-1} \end{gathered} Next, we substitute n= 9 into the equation and evaluate.
The ninth term is 378 125.
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A student sets the window of a graphing calculator to show an area of 96 square units.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"square units","answer":{"text":["98304"]}}
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Let's write the areas as a sequence and calculate the quotient of the consecutive terms to see if they are the same.
Since the quotient between both pairs of consecutive terms is 4, the sequence is a geometric sequence with the first term 96 and common ratio 4. Now we can write the explicit rule for this geometric sequence. a_n= 96( 4)^(n-1) We want to find the window's area after having zoomed out five times. Notice that zooming out five times corresponds to the sixth term of the sequence. Therefore, we will substitute n=6 into the formula and evaluate.
The screen area is 98 304 square units after zooming out five times.
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Geometric Sequences
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