57.1 Geometric Progression Problem Set
GEOMETRIC PROGRESSION | PRACTICE PROBLEMS Complete the following practice problems to reinforce your understanding of the concept covered in this module.
PROBLEM 1: The nth term of the noted geometric progression is best represented as: 243, −81, 27, …
A. −
* +
B. 243
243
. ,-. +
C. −243
. ,-. +
D. 243 −3
,-.
,-.
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PROBLEM 2: Determine the 3rd term in a geometric sequence if: 𝑎. = 8.1 𝑎2 = 240.1 A. 18.9 B. 192.1 C. 100 D. 44
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PROBLEM 3: The formula representing the 𝑛67 term of a geometric progression where 𝑟 = 2 and 𝑎. = 3 is best written as: A. 6, − 1 B. 5, C. 4, − 4 D. 7, − 2
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PROBLEM 4: Find 𝑡.* for a geometric progression where 𝑡. = 2 + 2𝑖 and 𝑟 = 3: A. 564,761 + 𝑖 B. 354,294 + 354,294𝑖 C. 77,714 + 7,715𝑖 D. 44 + 24𝑖
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PROBLEM 5: The sum of the series represented by the terms below is most close to: 1, 0.5, 0.25, 0.125, 0.0625, … A. 0.5 B. 1 C. 2 D. 4
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PROBLEM 6: The fourth term in the noted geometric progression is most close to: 243, −81, 27, … A. −398 B. 351 C. 3 D. – 9
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PROBLEM 7: The fifth term in the noted geometric progression is most close to: 243, −81, 27, … A. 3 B. −4 C. −1 D. 98
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PROBLEM 8: The 30th term in the noted geometric progression is most close to: 243, −81, 27, … A. −349,132 B. 351,987 C. −3.5 𝑥 10-.* D. 3.5 𝑥 10.*
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PROBLEM 9: Determine the 4th term in a geometric sequence if: 𝑎. = 8.1 𝑎2 = 240.1 A. −8.6 B. 192.1 C. 99.7 D. 102.5
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PROBLEM 10: The nth term of a geometric sequence with the variables defined below is best represented as: 𝑎. = 8.1 𝑎2 = 240.1 A. 8.1(2.33),-. B. 2.33(8.1),-. C. 240 2.33
,-.
D. 8.1(2.33),
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GEOMETRIC PROGRESSION | SOLUTIONS SOLUTION 1: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 243
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𝑎* = −81 𝑎+ = 27 The difference between each term is not UNIFORM, for example: 𝑎* − 𝑎. = −81 − 243 = −324 And: 𝑎+ − 𝑎* = 27 − (−81) = 108 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: 𝑎* −81 1 = =− 𝑎. 243 3 And: 𝑎+ 27 1 = =− 𝑎* −81 3
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This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as:
𝑟=−
1 3
𝑎. = 243 Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: 𝑎, = 𝑎. 𝑟 ,-. Plugging in the data we have defined up to this point we get: 1 𝑎, = 243 − 3
,-.
The correct answer choice is C. −𝟐𝟒𝟑
𝟏 𝒏-𝟏 𝟑
SOLUTION 2: The GENERAL FORMULA representing the 𝑛67 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.
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The value for the 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 8.1 𝑎2 = 240.1 Plugging these values in to our general formula, we get: 240.1 = 8.1𝑟 2-.
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Rearranging to isolate and solve for the COMMON RATIO, r, we get:
𝑟=
I
240.1 = 2.33 8.1
Finalizing our GENERAL FORMULA for this particular sequence, we have: 𝑎, = 8.1(2.33),-. To determine the 3rd term in this sequence, set: 𝑛=3 And we get: 𝑎+ = 8.1(2.33)+-. = 44 The correct answer choice is D. 𝟒𝟒
SOLUTION 3: The GENERAL FORMULA representing the 𝑛67 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.
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The value for the 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑟=2 𝑎. = 3 This becomes a simple plug and play problem with everything given to us in the problem statement. Taking our GENERAL FORMULA and plugging in our data, we get: 𝑎, = 3(2),-.
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This is not provided to us as an answer option, so we have to tweak it a bit to make it work. This formula as it stands right now, can be written as: 𝑎, = 6,-. The correct answer choice is A. 𝟔𝒏-𝟏
SOLUTION 4: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence
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Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑡. = 2 + 2𝑖 𝑟=3 In place of the standard 𝑎, and 𝑎. terminology, we are using 𝑡, and 𝑡. . Adjusting our GENERAL FORMULA, it will read as: 𝑡, = 𝑡. 𝑟 ,-. Plugging in the values that we have defined, we get: 𝑡, = (2 + 2𝑖)(3),-. To determine the 12th term in this sequence, set: 𝑛 = 12 And we get: 𝑡.* = (2 + 2𝑖)(3).*-.
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Which calculates out to: 𝑡.* = 2 + 2𝑖 177,147 Multiplying our terms through and simplifying, we get: 𝑡.* = 354,294 + 354,294𝑖 The correct answer choice is B. 𝟑𝟓𝟒, 𝟐𝟗𝟒 + 𝟑𝟓𝟒, 𝟐𝟗𝟒𝒊
SOLUTION 5: The GENERAL FORMULA highlighting the SUM of GEOMETRIC SERIES 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 SUM of a GEOMETRIC SEQUENCE is called a GEOMETRIC SERIES, and is expressed by the formula:
𝑎, 𝑎𝑟, 𝑎𝑟 * , 𝑎𝑟 + , … , 𝑎𝑟 ,-. = 𝑆, =
𝑎 1−𝑟
We have the first term of the sequence defined, which we can place in to this formula, but we need to determine the COMMON RATIO. To do that, let’s put some context around the GEOMETRIC PROGRESSION.
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The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 1 𝑎* = .5 𝑎+ = .25 𝑎O = .125 𝑎2 = .0625
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The difference between each term is not UNIFORM, for example: 𝑎O − 𝑎+ = .125 − .25 = −.125 And: 𝑎2 − 𝑎O = .0625 − .125 = −.0625 So with no COMMON DIFFERENCE, we can conclude that this is not an ARITHMETIC SEQUENCE. On the other hand, we do have a COMMON RATIO: 𝑎O . 125 = = .5 𝑎+ . 25 And: 𝑎2 . 0625 = = .5 𝑎O . 125 This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as: 𝑟 = .5 𝑎. = 1
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Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: 𝑎, = 𝑎. 𝑟 ,-. Plugging in the data we have defined up to this point we get: 𝑎, = 1 . 5
,-.
However, we are concerned in this case with the SUM of this GEOMETRIC SEQUENCE, expressed by the formula:
𝑆, =
𝑎 1−𝑟
We can plug and play in to the general formula, such that:
𝑆, =
1 =2 1 − .5
The correct answer choice is C. 𝟐
SOLUTION 6: The GENERAL FORMULA representing the 𝑛67 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.
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The value for the 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 243 𝑎* = −81 𝑎+ = 27 The difference between each term is not UNIFORM, for example: 𝑎* − 𝑎. = −81 − 243 = −324
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And: 𝑎+ − 𝑎* = 27 − (−81) = 108 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: 𝑎* −81 1 = =− 𝑎. 243 3 And: 𝑎+ 27 1 = =− 𝑎* −81 3 This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as:
𝑟=−
1 3
𝑎. = 243 Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: 𝑎, = 𝑎. 𝑟 ,-.
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Plugging in the data we have defined up to this point we get: 1 𝑎, = 243 − 3
,-.
To determine the 4th term in this sequence, set: 𝑛=4 And we get: 1 𝑎O = 243 − 3
O-.
= −9
The correct answer choice is D. – 𝟗
SOLUTION 7: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-.
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Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 243 𝑎* = −81 𝑎+ = 27 The difference between each term is not UNIFORM, for example: 𝑎* − 𝑎. = −81 − 243 = −324 And: 𝑎+ − 𝑎* = 27 − (−81) = 108 So with a no COMMON DIFFERENCE, we can conclude that this is not an ARITHMETIC SEQUENCE.
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On the other hand, we do have a COMMON RATIO: 𝑎* −81 1 = =− 𝑎. 243 3 And: 𝑎+ 27 1 = =− 𝑎* −81 3 This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as:
𝑟=−
1 3
𝑎. = 243 Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: 𝑎, = 𝑎. 𝑟 ,-. Plugging in the data we have defined up to this point we get: 1 𝑎, = 243 − 3
,-.
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To determine the 5th term in this sequence, set: 𝑛=5 And we get: 1 𝑎2 = 243 − 3
2-.
=3
The correct answer choice is A. 𝟑
SOLUTION 8: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence
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Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 243 𝑎* = −81 𝑎+ = 27 The difference between each term is not UNIFORM, for example: 𝑎* − 𝑎. = −81 − 243 = −324 And: 𝑎+ − 𝑎* = 27 − (−81) = 108 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: 𝑎* −81 1 = =− 𝑎. 243 3
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And: 𝑎+ 27 1 = =− 𝑎* −81 3 This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as:
𝑟=−
1 3
𝑎. = 243 Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: 𝑎, = 𝑎. 𝑟 ,-. Plugging in the data we have defined up to this point we get: 1 𝑎, = 243 − 3
,-.
To determine the 30th term in this sequence, set: 𝑛 = 30
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And we get:
𝑎+P
1 = 243 − 3
+P-.
= −3.5 𝑥 10-.*
The correct answer choice is A. – 𝟑. 𝟓 𝒙 𝟏𝟎-𝟏𝟐
SOLUTION 9: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence
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Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 8.1 𝑎2 = 240.1 Plugging these values in to our general formula, we get: 240.1 = 8.1𝑟 2-. Rearranging to isolate and solve for the COMMON RATIO, r, we get:
𝑟=
I
240.1 = 2.33 8.1
Finalizing our GENERAL FORMULA for this particular sequence, we have: 𝑎, = 8.1(2.33),-. To determine the 4th term in this sequence, set: 𝑛=4
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And we get: 𝑎+ = 8.1(2.33)O-. = 102.5 The correct answer choice is D. 𝟏𝟎𝟐. 𝟓
SOLUTION 10: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-.
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Let’s take a look at each of the terms we are given, we have: 𝑎. = 8.1 𝑎2 = 240.1 Plugging these values in to our general formula, we get: 240.1 = 8.1𝑟 2-. Rearranging to isolate and solve for the COMMON RATIO, r, we get:
𝑟=
I
240.1 = 2.33 8.1
Finalizing our GENERAL FORMULA for this particular sequence, we have: 𝑎, = 8.1(2.33),-. The correct answer choice is A. 𝟖. 𝟏(𝟐. 𝟑𝟑)𝒏-𝟏
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PROBLEM 1: The nth term of the noted geometric progression is best represented as: 243, −81, 27, …
A. −
* +
B. 243
243
. ,-. +
C. −243
. ,-. +
D. 243 −3
,-.
,-.
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PROBLEM 2: Determine the 3rd term in a geometric sequence if: 𝑎. = 8.1 𝑎2 = 240.1 A. 18.9 B. 192.1 C. 100 D. 44
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PROBLEM 3: The formula representing the 𝑛67 term of a geometric progression where 𝑟 = 2 and 𝑎. = 3 is best written as: A. 6, − 1 B. 5, C. 4, − 4 D. 7, − 2
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PROBLEM 4: Find 𝑡.* for a geometric progression where 𝑡. = 2 + 2𝑖 and 𝑟 = 3: A. 564,761 + 𝑖 B. 354,294 + 354,294𝑖 C. 77,714 + 7,715𝑖 D. 44 + 24𝑖
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PROBLEM 5: The sum of the series represented by the terms below is most close to: 1, 0.5, 0.25, 0.125, 0.0625, … A. 0.5 B. 1 C. 2 D. 4
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PROBLEM 6: The fourth term in the noted geometric progression is most close to: 243, −81, 27, … A. −398 B. 351 C. 3 D. – 9
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PROBLEM 7: The fifth term in the noted geometric progression is most close to: 243, −81, 27, … A. 3 B. −4 C. −1 D. 98
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PROBLEM 8: The 30th term in the noted geometric progression is most close to: 243, −81, 27, … A. −349,132 B. 351,987 C. −3.5 𝑥 10-.* D. 3.5 𝑥 10.*
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PROBLEM 9: Determine the 4th term in a geometric sequence if: 𝑎. = 8.1 𝑎2 = 240.1 A. −8.6 B. 192.1 C. 99.7 D. 102.5
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PROBLEM 10: The nth term of a geometric sequence with the variables defined below is best represented as: 𝑎. = 8.1 𝑎2 = 240.1 A. 8.1(2.33),-. B. 2.33(8.1),-. C. 240 2.33
,-.
D. 8.1(2.33),
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GEOMETRIC PROGRESSION | SOLUTIONS SOLUTION 1: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 243
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𝑎* = −81 𝑎+ = 27 The difference between each term is not UNIFORM, for example: 𝑎* − 𝑎. = −81 − 243 = −324 And: 𝑎+ − 𝑎* = 27 − (−81) = 108 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: 𝑎* −81 1 = =− 𝑎. 243 3 And: 𝑎+ 27 1 = =− 𝑎* −81 3
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This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as:
𝑟=−
1 3
𝑎. = 243 Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: 𝑎, = 𝑎. 𝑟 ,-. Plugging in the data we have defined up to this point we get: 1 𝑎, = 243 − 3
,-.
The correct answer choice is C. −𝟐𝟒𝟑
𝟏 𝒏-𝟏 𝟑
SOLUTION 2: The GENERAL FORMULA representing the 𝑛67 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.
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The value for the 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 8.1 𝑎2 = 240.1 Plugging these values in to our general formula, we get: 240.1 = 8.1𝑟 2-.
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Rearranging to isolate and solve for the COMMON RATIO, r, we get:
𝑟=
I
240.1 = 2.33 8.1
Finalizing our GENERAL FORMULA for this particular sequence, we have: 𝑎, = 8.1(2.33),-. To determine the 3rd term in this sequence, set: 𝑛=3 And we get: 𝑎+ = 8.1(2.33)+-. = 44 The correct answer choice is D. 𝟒𝟒
SOLUTION 3: The GENERAL FORMULA representing the 𝑛67 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.
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The value for the 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑟=2 𝑎. = 3 This becomes a simple plug and play problem with everything given to us in the problem statement. Taking our GENERAL FORMULA and plugging in our data, we get: 𝑎, = 3(2),-.
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This is not provided to us as an answer option, so we have to tweak it a bit to make it work. This formula as it stands right now, can be written as: 𝑎, = 6,-. The correct answer choice is A. 𝟔𝒏-𝟏
SOLUTION 4: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence
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Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑡. = 2 + 2𝑖 𝑟=3 In place of the standard 𝑎, and 𝑎. terminology, we are using 𝑡, and 𝑡. . Adjusting our GENERAL FORMULA, it will read as: 𝑡, = 𝑡. 𝑟 ,-. Plugging in the values that we have defined, we get: 𝑡, = (2 + 2𝑖)(3),-. To determine the 12th term in this sequence, set: 𝑛 = 12 And we get: 𝑡.* = (2 + 2𝑖)(3).*-.
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Which calculates out to: 𝑡.* = 2 + 2𝑖 177,147 Multiplying our terms through and simplifying, we get: 𝑡.* = 354,294 + 354,294𝑖 The correct answer choice is B. 𝟑𝟓𝟒, 𝟐𝟗𝟒 + 𝟑𝟓𝟒, 𝟐𝟗𝟒𝒊
SOLUTION 5: The GENERAL FORMULA highlighting the SUM of GEOMETRIC SERIES 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 SUM of a GEOMETRIC SEQUENCE is called a GEOMETRIC SERIES, and is expressed by the formula:
𝑎, 𝑎𝑟, 𝑎𝑟 * , 𝑎𝑟 + , … , 𝑎𝑟 ,-. = 𝑆, =
𝑎 1−𝑟
We have the first term of the sequence defined, which we can place in to this formula, but we need to determine the COMMON RATIO. To do that, let’s put some context around the GEOMETRIC PROGRESSION.
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The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 1 𝑎* = .5 𝑎+ = .25 𝑎O = .125 𝑎2 = .0625
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The difference between each term is not UNIFORM, for example: 𝑎O − 𝑎+ = .125 − .25 = −.125 And: 𝑎2 − 𝑎O = .0625 − .125 = −.0625 So with no COMMON DIFFERENCE, we can conclude that this is not an ARITHMETIC SEQUENCE. On the other hand, we do have a COMMON RATIO: 𝑎O . 125 = = .5 𝑎+ . 25 And: 𝑎2 . 0625 = = .5 𝑎O . 125 This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as: 𝑟 = .5 𝑎. = 1
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Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: 𝑎, = 𝑎. 𝑟 ,-. Plugging in the data we have defined up to this point we get: 𝑎, = 1 . 5
,-.
However, we are concerned in this case with the SUM of this GEOMETRIC SEQUENCE, expressed by the formula:
𝑆, =
𝑎 1−𝑟
We can plug and play in to the general formula, such that:
𝑆, =
1 =2 1 − .5
The correct answer choice is C. 𝟐
SOLUTION 6: The GENERAL FORMULA representing the 𝑛67 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.
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The value for the 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 243 𝑎* = −81 𝑎+ = 27 The difference between each term is not UNIFORM, for example: 𝑎* − 𝑎. = −81 − 243 = −324
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And: 𝑎+ − 𝑎* = 27 − (−81) = 108 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: 𝑎* −81 1 = =− 𝑎. 243 3 And: 𝑎+ 27 1 = =− 𝑎* −81 3 This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as:
𝑟=−
1 3
𝑎. = 243 Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: 𝑎, = 𝑎. 𝑟 ,-.
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Plugging in the data we have defined up to this point we get: 1 𝑎, = 243 − 3
,-.
To determine the 4th term in this sequence, set: 𝑛=4 And we get: 1 𝑎O = 243 − 3
O-.
= −9
The correct answer choice is D. – 𝟗
SOLUTION 7: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-.
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Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 243 𝑎* = −81 𝑎+ = 27 The difference between each term is not UNIFORM, for example: 𝑎* − 𝑎. = −81 − 243 = −324 And: 𝑎+ − 𝑎* = 27 − (−81) = 108 So with a no COMMON DIFFERENCE, we can conclude that this is not an ARITHMETIC SEQUENCE.
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On the other hand, we do have a COMMON RATIO: 𝑎* −81 1 = =− 𝑎. 243 3 And: 𝑎+ 27 1 = =− 𝑎* −81 3 This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as:
𝑟=−
1 3
𝑎. = 243 Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: 𝑎, = 𝑎. 𝑟 ,-. Plugging in the data we have defined up to this point we get: 1 𝑎, = 243 − 3
,-.
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To determine the 5th term in this sequence, set: 𝑛=5 And we get: 1 𝑎2 = 243 − 3
2-.
=3
The correct answer choice is A. 𝟑
SOLUTION 8: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence
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Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 243 𝑎* = −81 𝑎+ = 27 The difference between each term is not UNIFORM, for example: 𝑎* − 𝑎. = −81 − 243 = −324 And: 𝑎+ − 𝑎* = 27 − (−81) = 108 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: 𝑎* −81 1 = =− 𝑎. 243 3
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And: 𝑎+ 27 1 = =− 𝑎* −81 3 This confirms that we are working with a GEOMETRIC SEQUENCE with a COMMON RATIO and FIRST TERM defined as:
𝑟=−
1 3
𝑎. = 243 Referring back to our GENERAL FORMULA representing a GEOMETRIC SEQUENCE, we have: 𝑎, = 𝑎. 𝑟 ,-. Plugging in the data we have defined up to this point we get: 1 𝑎, = 243 − 3
,-.
To determine the 30th term in this sequence, set: 𝑛 = 30
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And we get:
𝑎+P
1 = 243 − 3
+P-.
= −3.5 𝑥 10-.*
The correct answer choice is A. – 𝟑. 𝟓 𝒙 𝟏𝟎-𝟏𝟐
SOLUTION 9: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence
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Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-. Let’s take a look at each of the terms we are given, we have: 𝑎. = 8.1 𝑎2 = 240.1 Plugging these values in to our general formula, we get: 240.1 = 8.1𝑟 2-. Rearranging to isolate and solve for the COMMON RATIO, r, we get:
𝑟=
I
240.1 = 2.33 8.1
Finalizing our GENERAL FORMULA for this particular sequence, we have: 𝑎, = 8.1(2.33),-. To determine the 4th term in this sequence, set: 𝑛=4
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And we get: 𝑎+ = 8.1(2.33)O-. = 102.5 The correct answer choice is D. 𝟏𝟎𝟐. 𝟓
SOLUTION 10: The GENERAL FORMULA representing the 𝑛67 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 𝑛67 term of a geometric progression, is represented by the expression: 𝑙 = 𝑎𝑟 ,-. Where: • 𝑙 is the value of the 𝑛67 term, which can otherwise be stated as 𝑎, • 𝑎 is the first term of the sequence, which can otherwise be stated as 𝑎. • 𝑟 is the COMMON RATIO • 𝑛 is the location of the term in the sequence Many times it’s easier, and often more familiar, to state this formula in the terms: 𝑎, = 𝑎. 𝑟 ,-.
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Let’s take a look at each of the terms we are given, we have: 𝑎. = 8.1 𝑎2 = 240.1 Plugging these values in to our general formula, we get: 240.1 = 8.1𝑟 2-. Rearranging to isolate and solve for the COMMON RATIO, r, we get:
𝑟=
I
240.1 = 2.33 8.1
Finalizing our GENERAL FORMULA for this particular sequence, we have: 𝑎, = 8.1(2.33),-. The correct answer choice is A. 𝟖. 𝟏(𝟐. 𝟑𝟑)𝒏-𝟏
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