56.1 Geometric Progression Concept Overview
GEOMETRIC PROGRESSION | CONCEPT OVERVIEW The topic of GEOMETRIC PROGRESSION can be referenced on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
CONCEPT INTRO: A GEOMETRIC PROGRESSION is a sequence of numbers, of which, each is obtained from the preceding number in the sequence by MULTIPLYING it by a certain fixed value called the COMMON RATIO. The pattern is that we are always taking the previous term and multiplying it by a fixed value to get to the next term in the sequence. A GEOMETRIC PROGRESSION can generally be written out as: π, ππ, ππ ! , ππ ! , β¦ , ππ !!! Where βπβ represents the COMMON RATIO and is raised to the power of one less the total number of terms being derived. To determine whether a given finite sequence is a GEOMETRIC PROGRESSION (G.P.), divide each term by the term preceding itβ¦of course, this can only be done with the terms after the first, since the first term has no term preceding it. If the quotients, or ratios, are equal, the sequence is considered to be GEOMETRIC.
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Not every progression, or sequence, that has a multiplicative pattern is geometric. It is considered a geometric sequence ONLY if it is being multiplied by the SAME number each and every time. The FORMULA to derive the π!! TERM of a GEOMETRIC PROGRESSION can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The value for the π!! term of a GEOMETRIC PROGRESSION, is represented by the expression: π = ππ !!! Where: β’ π is the value of the π!! term, which can otherwise be stated as π! β’ π is the first term of the sequence, which can otherwise be stated as π! β’ π is the COMMON RATIO β’ π is the location of the term in the sequence Many times itβs easier, and often more familiar, to state this formula in the terms: π! = π! π !!! The FORMULA for determining the SUM through the π!! TERM of a GEOMETRIC PROGRESSION can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Made with by Prepineer | Prepineer.com
The SUM up to the π!! term of a GEOMETRIC PROGRESSION, is represented by the expression: π(1 β π ! ) π= ;π β 1 (1 β π) Which can also be written as:
π=
(π β ππ) ;π β 1 (1 β π)
It is important to remember that the value of the COMMON RATIO must NOT BE EQUAL to 1. This would result in an undefined expression due to the zero term created in the DENOMINATOR. If the the COMMON RATIO is equal to 1, this tells us that each term of the GEOMETRIC PROGRESSION is equal, such that the SUM can be written as: π = π + π + π + β― = ππ The FIVE ELEMENTS of a GEOMETRIC PROGRESSION can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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The FIVE ELEMENTS of a GEOMETRIC PROGRESSION, and the letters by which they are denoted, are: 1. The first term is βπβ 2. The common difference is βπβ 3. The number of terms is βπβ 4. The last or βπ!! β term is βπβ 5. The sum of π terms is βπβ
CONVERGENCE OF A GEOMETRIC SERIES: The SUM of a GEOMETRIC SEQUENCE is called a GEOMETRIC SERIES, and is expressed by the formula:
π, ππ, ππ ! , ππ ! , β¦ , ππ !!! = π! =
π 1βπ
The GENERAL FORMULA used to define the LIMIT of a GEOMETRIC SEQUENCE can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. A GEOMETRIC SEQUENCE is said to DIVERGE, or be DIVERGENT, if a LIMIT does not exist, such that π > 1, otherwise expressed as:
lim π! β
!β!
π ;π > 1 1βπ
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A GEOMETRIC SEQUENCE is said to CONVERGE, or be CONVERGENT, if a LIMIT does exist, such that π < 1, otherwise expressed as:
lim π! =
!β!
π ;π < 1 (1 β π)
In the formula for a GEOMETRIC SEQUENCE, we can decrease the numerical value of the terms by increasing βπ", assuming the series converges, such that π < 1. By increasing βπβ, we can make the numerical value of the last term of π! smaller, and closer, to the LIMIT of the SEQUENCE.
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GEOMETRIC PROGRESSION | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material. The π!! term of the following geometric sequence is best represented as: ! !
, 1,2,4,8 β¦
A. 2 2
!!!
B.
!
2
!!!
C.
!
2
!!!
! !
D. 2
! !!! !
SOLUTION: The GENERAL FORMULA representing the π!! TERM of a GEOMETRIC PROGRESSION can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The value for the π!! term of an geometric progression, is represented by the expression: π = ππ !!!
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Where: β’ π is the value of the π!! term, which can otherwise be stated as π! β’ π is the first term of the sequence, which can otherwise be stated as π! β’ π is the COMMON RATIO β’ π is the location of the term in the sequence Many times itβs easier, and often more familiar, to state this formula in the terms: π! = π! π !!! Letβs take a look at each of the terms we are given, we have:
π! =
1 2
π! = 1 π! = 2 π! = 4 π! = 8 The difference between each term is not UNIFORM, for example: π! β π! = 2 β 1 = 1 And: π! β π! = 4 β 2 = 2
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So with a no COMMON DIFFERENCE, we can conclude that this is not an ARITHMETIC SEQUENCE. On the other hand, we do have a COMMON RATIO: π! 2 = =2 π! 1 And: π! 4 = =2 π! 2 This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as: π=2 π! =
1 2
Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: π! = π! π !!!
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Plugging in the data we have defined up to this point we get:
π! =
1 2 2
!!!
The correct answer choice is B.
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π π
π
π!π
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GEOMETRIC PROGRESSION | CONCEPT EXAMPLE Determine the tenth term of the following geometric sequence: ! !
, 1,2,4,8 β¦
A. 256 B. 541 C. 799 D. 265
SOLUTION: The GENERAL FORMULA representing the π!! TERM of a GEOMETRIC PROGRESSION can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The value for the π!! term of an geometric progression, is represented by the expression: π = ππ !!! Where: β’ π is the value of the π!! term, which can otherwise be stated as π! β’ π is the first term of the sequence, which can otherwise be stated as π!
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β’ π is the COMMON RATIO β’ π is the location of the term in the sequence Many times itβs easier, and often more familiar, to state this formula in the terms: π! = π! π !!! Letβs take a look at each of the terms we are given, we have:
π! =
1 2
π! = 1 π! = 2 π! = 4 π! = 8 The difference between each term is not UNIFORM, for example: π! β π! = 2 β 1 = 1 And: π! β π! = 4 β 2 = 2 So with a COMMON DIFFERENCE, we can conclude that is not an ARITHMETIC SEQUENCE.
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On the other hand, we do have a COMMON RATIO: π! 2 = =2 π! 1 And: π! 4 = =2 π! 2 This confirms that we are working with a GEOMETRIC SEQUENCE with COMMON RATIO and FIRST TERM defined as: π=2 !
π! = !. Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: π! = π! π !!! Plugging in the data we have defined up to this point we get:
π! =
1 2 2
!!!
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To determine the value of the 10!! term of this GEOMETRIC SEQUENCE, we need to plug in: π = 10 Doing this, we get:
π!" =
1 2 2
!"!!
= 256
The correct answer choice is A. 256
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CONCEPT INTRO: A GEOMETRIC PROGRESSION is a sequence of numbers, of which, each is obtained from the preceding number in the sequence by MULTIPLYING it by a certain fixed value called the COMMON RATIO. The pattern is that we are always taking the previous term and multiplying it by a fixed value to get to the next term in the sequence. A GEOMETRIC PROGRESSION can generally be written out as: π, ππ, ππ ! , ππ ! , β¦ , ππ !!! Where βπβ represents the COMMON RATIO and is raised to the power of one less the total number of terms being derived. To determine whether a given finite sequence is a GEOMETRIC PROGRESSION (G.P.), divide each term by the term preceding itβ¦of course, this can only be done with the terms after the first, since the first term has no term preceding it. If the quotients, or ratios, are equal, the sequence is considered to be GEOMETRIC.
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by Prepineer | Prepineer.com
Not every progression, or sequence, that has a multiplicative pattern is geometric. It is considered a geometric sequence ONLY if it is being multiplied by the SAME number each and every time. The FORMULA to derive the π!! TERM of a GEOMETRIC PROGRESSION can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The value for the π!! term of a GEOMETRIC PROGRESSION, is represented by the expression: π = ππ !!! Where: β’ π is the value of the π!! term, which can otherwise be stated as π! β’ π is the first term of the sequence, which can otherwise be stated as π! β’ π is the COMMON RATIO β’ π is the location of the term in the sequence Many times itβs easier, and often more familiar, to state this formula in the terms: π! = π! π !!! The FORMULA for determining the SUM through the π!! TERM of a GEOMETRIC PROGRESSION can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Made with by Prepineer | Prepineer.com
The SUM up to the π!! term of a GEOMETRIC PROGRESSION, is represented by the expression: π(1 β π ! ) π= ;π β 1 (1 β π) Which can also be written as:
π=
(π β ππ) ;π β 1 (1 β π)
It is important to remember that the value of the COMMON RATIO must NOT BE EQUAL to 1. This would result in an undefined expression due to the zero term created in the DENOMINATOR. If the the COMMON RATIO is equal to 1, this tells us that each term of the GEOMETRIC PROGRESSION is equal, such that the SUM can be written as: π = π + π + π + β― = ππ The FIVE ELEMENTS of a GEOMETRIC PROGRESSION can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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The FIVE ELEMENTS of a GEOMETRIC PROGRESSION, and the letters by which they are denoted, are: 1. The first term is βπβ 2. The common difference is βπβ 3. The number of terms is βπβ 4. The last or βπ!! β term is βπβ 5. The sum of π terms is βπβ
CONVERGENCE OF A GEOMETRIC SERIES: The SUM of a GEOMETRIC SEQUENCE is called a GEOMETRIC SERIES, and is expressed by the formula:
π, ππ, ππ ! , ππ ! , β¦ , ππ !!! = π! =
π 1βπ
The GENERAL FORMULA used to define the LIMIT of a GEOMETRIC SEQUENCE can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. A GEOMETRIC SEQUENCE is said to DIVERGE, or be DIVERGENT, if a LIMIT does not exist, such that π > 1, otherwise expressed as:
lim π! β
!β!
π ;π > 1 1βπ
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A GEOMETRIC SEQUENCE is said to CONVERGE, or be CONVERGENT, if a LIMIT does exist, such that π < 1, otherwise expressed as:
lim π! =
!β!
π ;π < 1 (1 β π)
In the formula for a GEOMETRIC SEQUENCE, we can decrease the numerical value of the terms by increasing βπ", assuming the series converges, such that π < 1. By increasing βπβ, we can make the numerical value of the last term of π! smaller, and closer, to the LIMIT of the SEQUENCE.
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by Prepineer | Prepineer.com
GEOMETRIC PROGRESSION | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material. The π!! term of the following geometric sequence is best represented as: ! !
, 1,2,4,8 β¦
A. 2 2
!!!
B.
!
2
!!!
C.
!
2
!!!
! !
D. 2
! !!! !
SOLUTION: The GENERAL FORMULA representing the π!! TERM of a GEOMETRIC PROGRESSION can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The value for the π!! term of an geometric progression, is represented by the expression: π = ππ !!!
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Where: β’ π is the value of the π!! term, which can otherwise be stated as π! β’ π is the first term of the sequence, which can otherwise be stated as π! β’ π is the COMMON RATIO β’ π is the location of the term in the sequence Many times itβs easier, and often more familiar, to state this formula in the terms: π! = π! π !!! Letβs take a look at each of the terms we are given, we have:
π! =
1 2
π! = 1 π! = 2 π! = 4 π! = 8 The difference between each term is not UNIFORM, for example: π! β π! = 2 β 1 = 1 And: π! β π! = 4 β 2 = 2
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So with a no COMMON DIFFERENCE, we can conclude that this is not an ARITHMETIC SEQUENCE. On the other hand, we do have a COMMON RATIO: π! 2 = =2 π! 1 And: π! 4 = =2 π! 2 This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as: π=2 π! =
1 2
Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: π! = π! π !!!
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Plugging in the data we have defined up to this point we get:
π! =
1 2 2
!!!
The correct answer choice is B.
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π π
π
π!π
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GEOMETRIC PROGRESSION | CONCEPT EXAMPLE Determine the tenth term of the following geometric sequence: ! !
, 1,2,4,8 β¦
A. 256 B. 541 C. 799 D. 265
SOLUTION: The GENERAL FORMULA representing the π!! TERM of a GEOMETRIC PROGRESSION can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The value for the π!! term of an geometric progression, is represented by the expression: π = ππ !!! Where: β’ π is the value of the π!! term, which can otherwise be stated as π! β’ π is the first term of the sequence, which can otherwise be stated as π!
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β’ π is the COMMON RATIO β’ π is the location of the term in the sequence Many times itβs easier, and often more familiar, to state this formula in the terms: π! = π! π !!! Letβs take a look at each of the terms we are given, we have:
π! =
1 2
π! = 1 π! = 2 π! = 4 π! = 8 The difference between each term is not UNIFORM, for example: π! β π! = 2 β 1 = 1 And: π! β π! = 4 β 2 = 2 So with a COMMON DIFFERENCE, we can conclude that is not an ARITHMETIC SEQUENCE.
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On the other hand, we do have a COMMON RATIO: π! 2 = =2 π! 1 And: π! 4 = =2 π! 2 This confirms that we are working with a GEOMETRIC SEQUENCE with COMMON RATIO and FIRST TERM defined as: π=2 !
π! = !. Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: π! = π! π !!! Plugging in the data we have defined up to this point we get:
π! =
1 2 2
!!!
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To determine the value of the 10!! term of this GEOMETRIC SEQUENCE, we need to plug in: π = 10 Doing this, we get:
π!" =
1 2 2
!"!!
= 256
The correct answer choice is A. 256
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