UNIT III STUDY GUIDE
Number Theory and the
Real Number System
Course Learning Outcomes for Unit III
Upon completion of this unit, students should be able to:
3. Perform computations involving exponents, scientific notations, and sequences within the real
number system.
3.1 Identify irrational numbers.
3.2 Apply basic rules for exponents and radicals to solve problems.
3.3 Perform operations using scientific notation.
3.4 Compute future terms of arithmetic and geometric sequences.
3.5 Apply Fibonacci sequence principles to solve problems.
Reading Assignment
Chapter 6:
Number Theory and the Real Number System: Understanding the Numbers All Around Us
Section 6.4: The Real Number System, pp. 265-274
Section 6.5: Exponents and Scientific Notation, pp. 275-284
Section 6.6: Looking Deeper: Sequences, pp. 285-294
Unit Lesson
Chapter 6: Number Theory and the Real Number System
Unit I introduced counting numbers, integers, and rational numbers. In this section, you will learn about other
types of numbers that make up the real number system.
6.4 The Real Number System
Mathematicians have categorized numbers into different sets. As shown previously, some sets of numbers
have different rules when performing computations. For example, integer operations depend on the sign of
the number and fraction operations require that we manipulate the denominator when adding or subtracting.
Irrational Numbers
Recall that rational numbers are numbers that can be expressed as a fraction. For example, whole numbers,
integers, and some decimals are rational numbers.
Irrational numbers are numbers that are not rational and cannot be expressed as a fraction. More
specifically, irrational numbers are numbers that when written as a decimal do not repeat and do not
terminate. Their decimal expansion never ends and there is no pattern to the expansion. For example,
312.458967231547485693215…. is an irrational number.
Example:
Is the following a rational number or irrational number?
3
8
This is a rational number because it is expressed as a fraction.
MAT 1301, Liberal Arts Math
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Example:
UNIT x STUDY GUIDE
Is the following a rational number or is it an irrational number?
Title
1.23456789101112 …
This is an irrational number because the decimal does NOT terminate or repeat.
Example:
Is the following a rational number or is it an irrational number?
0.101101110…
This is an irrational number because the decimal does NOT terminate or repeat.
Irrational numbers are identified by mathematical symbols because they are impossible to write otherwise.
For example, the numbers represented by 𝜋, 𝑒, or 𝜑 are irrational. These numbers have special symbols
because they reoccur in several mathematical areas such as the study of geometry.
Computing Radicals
The radical,√ , provides another way of expressing an irrational number. A radical is sometimes called a
root. This symbol is found on your calculator. An example of an irrational number expressed as a radical
is √2. The diagram below provides some key terms of a radical. We will need to know these terms when
performing calculations with radicals.
Example:
Is the following a rational number or irrational number?
√81
Solution:
We need to solve the radical to determine if the √81 is rational or irrational. The radical sign undoes a perfect
square. Therefore, the radical can be simplified if the radicand has two factors that are the same.
We can also find the √81 by using our calculator.
Answer: The √81 is a rational number, because √81 = 9 and 9 is rational number.
We will discuss how to multiply, divide, and add radicals. To do this, we will need to know the following
property.
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Multiplying and Dividing Radicals – If a ≥ 0 and b ≥ 0,
UNIT x STUDY GUIDE
Title
Example:
Simplify:
√27
Solution:
Rewrite the radicand as two factors (9 • 3). We know that √9 = √3 ∙ 3 = 3.
Therefore, we have
Example:
Simplify:
√189
Solution:
Rewrite the radicand as two factors (9 • 21). We know that √9 = √3 ∙ 3 = 3. Therefore, we have
Example:
Perform the indicated operation, and simplify if possible:
√54
√6
Solution:
Example:
Perform the indicated operation, and simplify if possible:
√12√15
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Solution:
UNIT x STUDY GUIDE
Title
√12√15
= √12 ∙ 15
Rewrite the expression so that both numbers are
underneath the radical.
= √4 ∙ 3 ∙ 3 ∙ 5 Factor 12 and 15. Notice that 12 = 4 • 3 and 15 = 3 • 5
= √4√3 ∙ 3 ∙ 5 We know that √4 = 2. Therefore, we separate this.
= 2√9 ∙ 5
Place a 2 on the outside of radical because √4 = 2.
= 2√9√5
Separate the radicals.
= 2 ∙ 3√5
We know that √9 = 3. Move the 3 to the outside.
= 6√5
Multiply the numbers that are outside the radical. The √5
is irrational and cannot be simplified.
Our answer is: 𝟔√𝟓.
Rationalizing Denominators
Recall that the denominator is the bottom number of a fraction and that a fraction bar represents division. In
this section, we will discuss the process of rationalizing the denominator. This means that we will rewrite the
quotient or fraction so that it has no radicals in the denominator.
Review the following example to learn the process for rationalizing the denominator.
Example:
Rationalize the denominator and simplify:
4
√3
Solution:
1. Multiply the numerator and denominator of the fraction by the radical that is in the denominator.
4
√3
=
4 ∙ √𝟑
√3 ∙ √𝟑
2. Simplify the numerator and denominator by performing the indicated operations.
4 ∙ √3
√3 ∙ √3
=
4√3
√3 ∙ 3
3. Simplify the denominator.
4√3
√3 ∙ 3
MAT 1301, Liberal Arts Math
=
4√3
√9
=
𝟒√𝟑
𝟑
4
Adding Expressions Containing Radicals
UNIT x STUDY GUIDE
Title
The radicands of a radical expression must be the same in order to add or subtract them. For example, √3 +
√2 cannot be added because the radicands are different. If the radicands are the same, then we keep the
radicand and add the numbers outside the radical together. For example, 3√2 + 4√2 = 7√2.
Example:
If possible, combine the radicals into a single radical:
2√3 − 4√2
This is NOT possible. The radicands are different so we cannot simplify this expression any further.
We must first simplify the radical to determine if two radicals can be added or subtracted. If the simplified
radicals do not have the same radicand, then we cannot perform the operation.
Example:
If possible, combine the radicals into a single radical:
√20 + 6√5
Solution:
Simplify each radical. The √5 cannot be simplified any further, so we will simplify √20.
Properties of Real Numbers
The diagram below represents the set of real numbers. As shown, there are two distinct sets of real numbers:
rational and irrational numbers. Rational numbers are comprised of natural numbers, whole numbers, and
integers. It is also shown that the set of whole numbers includes the set of natural numbers, and the set of
integers includes the set of whole and natural numbers.
Chart depicting set of real numbers (Pirnot, 2014, p. 270)
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While different, each of the sets in the diagram follows guidelines. These guidelines
calledGUIDE
the properties
UNIT xare
STUDY
of real numbers. The properties are listed on page 271 of your textbook.
Title
Example:
State which property of the real numbers we are illustrating:
3(4 + 5) = 3 ∙ 4 + 3 ∙ 5
Distributive: The 3 is being multiplied by both numbers in the parentheses.
Example:
State which property of the real numbers we are illustrating:
3 + (6 + 8) = (6 + 8) + 3
Commutative property for addition: The order in which we add changed.
Example:
State which property of the real numbers we are illustrating:
3 + (6 + 8) = (3 + 6) + 8
Associative property for addition: The way we grouped the numbers in the parentheses changed.
Example:
State which property of the real numbers we are illustrating:
7+0= 7
Identity element for addition: Adding zero does not change the original number.
Example:
State which property of the real numbers we are illustrating:
8 + (−8) = 0
Additive inverse: We added opposite numbers together.
6.5 Exponents and Scientific Notation: Exponential and scientific notation is used to identify very small and
very large numbers. For example, these notations will help us to express a number that is 100 digits long.
Exponents and scientific notation will also aid in performing calculations with very small and very large
numbers.
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Exponents
UNIT x STUDY GUIDE
Title
Exponential notation provides a way of identifying a large or small number. The diagram below represents the
basic definition of an exponent operation.
As shown, the exponent identifies the number of times the base is being multiplied to itself. For example, 46 =
4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 and 422 = 42 ∙ 42. We can perform the multiplication to find the answer or use the exponent
key on our calculator. This key is usually represented by the “^” or “𝑥 𝑦 ” symbol on your calculator.
Example:
Evaluate: 53
Solution:
Example:
Evaluate:−32
Solution:
Example:
Evaluate: (−4)4
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Solution:
UNIT x STUDY GUIDE
Title
Rules for Exponents
There are several rules that can be applied to exponents. These rules will help us when performing
mathematical operations. First, we will discuss the product rule for exponents.
Product Rule for Exponents – If x is a real number and m and n are natural numbers, then
𝑥 𝑚 𝑥 𝑛 = 𝑥 𝑚+𝑛
Note: We can only apply this rule if both bases are the same.
Example:
Evaluate: 32 ∙ 34
Solution:
Each exponential expression has a base of 3. Therefore, we can apply the product rule for exponents. To do
this, keep the base and add the exponents together. Then, expand the exponential problem to find the
answer:
32 ∙ 34 = 32+4 = 36 = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 = 𝟕𝟐𝟗
The power is applied when an exponential expression is raised to a power or exponent. In this case, we will
multiply the exponents together and keep the base.
Power Rule for Exponents – If x is a real number and m and n are natural numbers, then
(𝑥 𝑚 ) 𝑛 = 𝑥 𝑚∙𝑛
Example:
Evaluate: (72 )3
Solution:
Apply the power rule for exponents to solve. To do this, multiply the exponents together and keep the base.
Then, expand the exponential expression to find the answer:
(72 )3 = 72∙3 = 76 = 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 = 𝟏𝟏𝟕, 𝟔𝟒𝟗
Another rule we will discuss is the quotient rule for exponents. This rule is applied when we are dividing by
two exponential expressions.
Example:
Evaluate: (7−1 )−3
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Solution:
UNIT x STUDY GUIDE
Title
Apply the power rule to solve. To do this, multiply the exponents together and keep the base. Then, expand
the exponential expression. Note: A negative number times a negative number is a positive number.
(7−1 )−3 = 7−1(−3) = 73 = 7 ∙ 7 ∙ 7 = 343
Quotient Rule for Exponents – If x is a nonzero real number and both m and n are natural numbers, then
𝑥𝑚
= 𝑥 𝑚−𝑛
𝑥𝑛
Note: We can only apply this rule if both bases are the same.
Example:
Evaluate:
59
57
Solution:
Apply the quotient rule for exponents to solve. To do this, subtract the exponent attached to the denominator
from the exponent attached to the numerator and keep the base. Then, expand the exponential problem to
find the answer:
59
= 59−7 = 52 = 5 ∙ 5 = 𝟐𝟓
57
Special Exponent Definitions
Exponential expressions have special definitions when they are raised to a zero or negative power.
Example:
Evaluate: 30
Solution:
Definition 1 above states that any number raised to the zero power is 1. So,
30 = 𝟏
Example:
Evaluate: 5−2
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Solution:
UNIT x STUDY GUIDE
Title
Definition 2 above states to make the exponent positive by placing a 1 over the exponential expression. So,
5−2 =
1
1
𝟏
=
=
52 5 ∙ 5 𝟐𝟓
Next, we will combine rules and definitions to solve more complex exponential problems in the following
examples.
Example:
Evaluate: 54 ∙ 5−6
Solution:
First, apply the product rule for exponents. (Keep the base and add the exponents.)
54 ∙ 5−6 = 54+(−6) = 5−2
The first step resulted in a negative exponent. Therefore, apply the definition for negative exponents. (Make
the exponent positive and place a 1 over the exponential expression.)
=
1
1
𝟏
=
=
52 5 ∙ 5 𝟐𝟓
Example:
(−3)6
Evaluate: (−3)9
Solution:
First, apply the quotient rule for exponents. (Subtract the exponent attached to the denominator from the
exponent attached to the numerator and keep the base.)
(−3)6
= (−3)6−9 = (−3)−3
(−3)9
The first step resulted in a negative exponent. Therefore, apply the definition for negative exponents. (Make
the exponent positive and place a 1 over the exponential expression.)
=
1
1
1
𝟏
=
=
=−
3
(−3)
−3 ∙ −3 ∙ −3 −27
𝟐𝟕
Example:
Evaluate: (3−4 )0
Solution:
Any number raised to the zero power is 1. So,
(3−4 )0 = 3−4(0) = 30 = 𝟏
Scientific Notation
Scientific notation is used when representing very small or very large numbers. The form for scientific notation
is listed below. Notice that this form includes an exponential expression whose base is always 10.
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Scientific notation – A number is written in scientific notation if it is in the form UNIT x STUDY GUIDE
𝑎 × 10𝑛
Title
Where 1 ≤ 𝑎 ≤ 10, and n is any integer.
The following chart is listed on page 280 of your textbook. It reviews the steps for converting a number into
scientific notation.
Chart containing rules for converting a decimal number to scientific notation
(Pirnot, 2014, p. 280)
Example:
Rewrite 3,200,000,000 in scientific notation.
Solution:
This number is very large, so we will move the decimal to the left until there is only one number in front of the
decimal. To do this, we will move the decimal nine places to the left. Therefore, our exponent is 9.
3,200,000,000 = 𝟑. 𝟐 × 𝟏𝟎𝟗
Example:
Rewrite 0.000258 in scientific notation.
Solution:
This number is very small, so we will move the decimal to the right until there is one number in front of the
decimal. To do this, we will move the decimal four places to the right. Therefore, our exponent is – 4.
0.000258 = 𝟐. 𝟓𝟖 × 𝟏𝟎−𝟒
The next examples practice converting scientific notation to a decimal.
Example:
Rewrite 1.78 × 10−3 in standard notation.
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Solution:
UNIT x STUDY GUIDE
Title
The exponent is negative. This means that our number will be small. Therefore, we will move the decimal 3
places to the left.
1.78 × 10−3 = 0.00178
Multiplying and Dividing Numbers in Scientific Notation
We can multiply and divide numbers that are written in scientific notation by following the rules for exponents.
The examples shown below discuss how to do this.
Example:
Use scientific notation to perform the following operation. Leave your answer in scientific notation form.
(4 × 103 )(2 × 104 )
Solution:
First, separate the numbers and exponential expressions.
(4 × 103 )(2 × 104 ) = (4 ∙ 2) × (103 ∙ 104 )
Next, multiply each set. Use the product rule for exponents to multiply the exponential expressions.
= 8.0 × 103+4 = 𝟖. 𝟎 × 𝟏𝟎𝟕
Example:
Use scientific notation to perform the following operation. Leave your answer in scientific notation form.
(8 × 10−2 ) ÷ (2 × 103 )
Solution:
First, write the division problem as a fraction:
(8 × 10−2 ) ÷ (2 × 103 ) =
8 × 10−2
2 × 103
Next, separate the numbers and exponential expressions:
=
8 10−2
×
2 103
Divide each set. Use the quotient rule for exponents to divide the exponential expressions.
= 4 × 10−2−3 = 4 × 10−2+(−3) = 𝟒 × 𝟏𝟎−𝟓
Example:
Use scientific notation to perform the following operation. Leave your answer in scientific notation form.
(9.6368 × 103 )(4.15 × 10−6 )
1.52 × 104
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Solution:
UNIT x STUDY GUIDE
First, separate the numbers and exponential expressions.
Title
(9.6368 × 103 )(4.15 × 10−6 ) 9.6368 ∙ 4.15 103 ∙ 10−6
=
×
1.52 × 104
1.52
104
Next, multiply the numbers or expressions in the numerator.
=
39.99272 103+(−6)
×
1.52
104
Divide both sets. Use the quotient rule for exponents to divide the exponential expressions.
= 26.311 ×
10−3
= 26.311 × 10−3−4 = 26.311 × 10−3+(−4) = 26.311 × 10−7
104
Move the decimal one more place to the left. This is our answer.
= 𝟐. 𝟔𝟑𝟏𝟏 × 𝟏𝟎−𝟔
Applications of Scientific Notation
Scientific notation is used in real-world applications. The next problem provides an example.
Example:
In 1977, scientists sent the spaceship Voyager II to Neptune, which is about 2.8 billion miles away. If the
spacecraft averaged about 25000 miles per hour, how long did the journey take?
Solution:
The numbers presented in this problem are very large. Therefore, we will use scientific notation to solve.
𝐼𝑛 𝑠𝑐𝑖𝑒𝑛𝑡𝑖𝑓𝑖𝑐 𝑛𝑜𝑡𝑎𝑡𝑖𝑜𝑛, 2.8 𝑏𝑖𝑙𝑙𝑖𝑜𝑛 𝑚𝑖𝑙𝑒𝑠 = 2.8 × 109
𝐼𝑛 𝑠𝑐𝑖𝑒𝑛𝑡𝑖𝑓𝑖𝑐 𝑛𝑜𝑡𝑎𝑡𝑖𝑜𝑛, 25,000 𝑚𝑖𝑙𝑒𝑠 𝑝𝑒𝑟 ℎ𝑜𝑢𝑟 = 2.5 × 104
Next, divide the numbers of miles by the number of miles per hour to find how long the journey took.
2.8 × 109 𝑚𝑖𝑙𝑒𝑠 2.8 109
=
×
= 1.12 × 109−4 = 𝟏. 𝟏𝟐 × 𝟏𝟎𝟓 𝒉𝒐𝒖𝒓𝒔
𝑚𝑖𝑙𝑒𝑠 2.5 104
4
2.5 × 10
ℎ𝑜𝑢𝑟
For more information regarding scientific notation and its use, please view the following interactive
presentation.
https://media.pearsoncmg.com/pcp/pls/pls_mycoursetools/fufillment/mct_1256689785_csu/trigonometry/redir
ect_math_custom_index_20.html
6.6 Sequences: As a child, we were taught to recognize patterns. We might have been given three shapes
and were asked to provide the fourth shape based upon a recognizable pattern. As adults, we can recognize
patterns or trends in things such as utility bills or gas prices. For example, we may expect an increase in our
cable bill if that bill increased in previous years. In this section, we will learn how to recognize patterns in
number sequences and predict the next number in a given sequence.
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Sequence - A sequence is a list of numbers that follows some rule or pattern. UNIT
The numbers
the list are
x STUDYinGUIDE
called terms of the sequence. A sequence is in the form:
Title
We will discuss three sequences: arithmetic, geometric, and the Fibonacci sequence.
Arithmetic Sequence
An arithmetic sequence is a sequence in which the future terms are found by subtracting or adding a fixed
constant to a previous term. The fixed constant is referred to as the common difference. For example, the
sequence 2, 8, 14, 20…. Is an arithmetic sequence because each term is found by adding 6 to the previous
term. The common difference for this example is 6.
Example:
List the next two terms of the arithmetic sequence:
11, 7, 3, -1, …
Solution:
The first term of the sequence is 11. To proceed, we need to determine what operation was done to result in a
7. We know that 11 – 4 = 7. Let’s see if this rule works for the other numbers in the sequence:
11 – 4 = 7
7–4=3
3 – 4 = -1
Subtracting 4 from the previous terms in the sequence results in the next consecutive term. Therefore, the
common difference is – 4. We will find the next two terms by following this pattern:
-1 – 4 = -5
-5 – 4 = -9
The next two terms are -5 and -9.
We will be given certain criteria about an arithmetic sequence and will be asked to find the nth term of that
sequence. To do this, we need to know the common difference (d), the first term (𝑎1 ), and the number of
terms in the sequence (n).
The nth Term of an Arithmetic Sequence – The nth term of an arithmetic sequence with first term 𝑎1 and
common difference d is
𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 .
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We can find the sum of first n terms of an arithmetic sequence once the nth term(
𝑎𝑛x) of
the sequence
UNIT
STUDY
GUIDE is
found.
Title
The Sum of the First n Terms of Arithmetic Sequence – The sum of the first n terms of an arithmetic
sequence is given by
𝑛(𝑎1 + 𝑎𝑛 )
2
Example:
For the arithmetic sequence, find the specified term 𝑎𝑛 and then find the sum of the terms from 𝑎1 to 𝑎𝑛 ,
inclusive.
11, 17, 23, 29… Find 𝑎9
Solution:
Use the following formula to solve for 𝒂𝟗 :
𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
First, we need to identify 𝑎1 , n, and d of the sequence.
11, 17, 23, 29…
𝑎1 = first term = 11
Next, plug in 𝑎1 = 11, d = 6, and n = 9 into the formula:
𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
𝑎9 = 11 + (9 − 1) ∙ 6
= 11 + 8 ∙ 6
= 11 + 48
= 𝟓𝟗
Therefore, 𝒂𝟗 = 𝟓𝟗.
Now we will find the sum of the terms from 𝑎1 to 𝑎𝑛 . To do this, we will use the value we found for 𝑎9 and
solve the following formula:
𝑛(𝑎1 + 𝑎𝑛 )
2
Recall that a1 = 11, a9 = 59, and n = 9. Therefore,
𝑛(𝑎1 + 𝑎𝑛 ) 9(11 + 59) 9 ∙ 70
=
=
= 9 ∙ 35 = 𝟑𝟏𝟓
2
2
2
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The sum of the terms from 𝒂𝟏 to 𝒂𝟗 is 315.
UNIT x STUDY GUIDE
Title
Geometric Sequence
A geometric sequence is a sequence in which the future terms are found by multiplying a fixed constant to a
previous term. The fixed constant is referred to as a common ratio. For example, the sequence 3, 9, 27, 81….
Is a geometric sequence because each term is found by multiplying 3 to the previous term. The common ratio
for this example is 3.
Example:
List the next two terms of the sequence:
8, 24, 72, 216, …
Solution:
The first term of the sequence is 8. To proceed, we need to determine what operation was done to result in a
24. We know that 8 • 3 = 24. Let’s see if this rule works for the other numbers in the sequence:
8 • 3 = 24
24 • 3 = 72
72 • 3 = 216
Multiplying 3 by the previous terms in the sequence results in the next consecutive term. Therefore, the
common ratio is 3. We will find the next two terms by following this pattern:
216 • 3 = 648
648 • 3 = 1,944
The next two terms are 648 and 1,944.
We will be given certain criteria about a geometric sequence and will be asked to find the nth term of that
sequence. To do this, we need to know the common ratio ®, the first term (𝑎1 ), and the number of terms in the
sequence (n).
The nth Term of a Geometric Sequence – The nth term of a geometric sequence with a common ratio r is
𝑎𝑛 = 𝑎1 ∙ 𝑟 𝑛−1 .
Example:
For the geometric sequence for the indicated term:
2, -4, 8, -16, …Find 𝑎10 .
Solution:
We will use the following formula to solve for 𝒂𝟏𝟎 :
𝑎𝑛 = 𝑎1 ∙ 𝑟 𝑛−1
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First, identify 𝑎1 , n , and r of the sequence:
UNIT x STUDY GUIDE
Title
Next, plug in 𝑎1 = 2, r = -2, and n = 10 into the formula:
10−1
𝑎10 = 2 ∙ (−2)
𝑎𝑛 = 𝑎1 ∙ 𝑟 𝑛−1
= 2 ∙ (−2)9 = 2 ∙ (−512) = −𝟏, 𝟎𝟐𝟒
Therefore, 𝑎10 = −1,024.
The Fibonacci Sequence
The Fibonacci Sequence is a special sequence that was discovered in 1202 to aid in numeric calculations.
The Fibonacci Sequence – A sequence whose future terms is the sum of the two previous terms in the
sequence:
1,1,2,3,5,8,13,21,34,55,89,….
Note: we label the terms of the sequence F1, F2, F3, F4…
Example:
Two terms are given in the Fibonacci sequence. Find the specified term.
F11 = 89 and F13 = 233. Find F12.
Solution:
A term of the Fibonacci sequence is found by adding the two previous terms together. Assume that we are
given the 11th, 12th, and 13th term of the sequence. This means that the 13th term would be found by adding
the 11th and 12th terms together. This is represented by the formula
𝐹13 = 𝐹11 + 𝐹12
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We will plug in the values for 𝐹11 and 𝐹13and solve for 𝐹12.
UNIT x STUDY GUIDE
Title
For more information about sequencing, please view the following interactive presentation:
https://media.pearsoncmg.com/pcp/pls/pls_mycoursetools/fufillment/mct_1256689785_csu/basic_math/redire
ct_math_custom_index_64.html
Reference
Pirnot, T. L. (2014). Mathematics all around (5th ed.). Boston, MA: Pearson.
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