55.1 Arithmetic Progression Problem Set

ARITHMETIC PROGRESSION | PRACTICE PROBLEMS Complete the following to reinforce your understanding of the concept covered in this module.

PROBLEM 1: The 𝑛"# term of an arithmetic sequence with a 4"# term equal to 65 and an 8"# term equal to 93 is best represented as: A. 110 − 5 𝑛 − 1 B. 110 + 5 𝑛 + 1 C. 44 + 7 𝑛 − 1 D. 44 − 7 𝑛 − 1



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PROBLEM 2: Determine the 𝑛"# term of an arithmetic sequence given: 𝑎1 = 5 𝑎34 = 159 A. −83 + 22 𝑛 − 1 B. −83 + 22 𝑛 + 1 C. 83 + 22 𝑛 − 1 D. 83 − 22 𝑛 − 1



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PROBLEM 3: The value of “𝑛” for which the following summation holds true is most close to: 8

0.25𝑖 + 2 = 21 9:3

A. 7 B. 5 C. 10 D. 22



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PROBLEM 4: If the first term of an arithmetic sequence is equal to 6 and the common difference is equal to 3, the value of the 50"# term is most close to: A. 13 B. 41 C. 35 D. 153



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PROBLEM 5: The sum of all the positive integers from 5 to 1,555 that are divisible by 5 is most close to: A. 187,629 B. 242,580 C. 917,444 D. 346,745



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PROBLEM 6: The second term of an arithmetic sequence with a 4"# term value of 65 and an 8"# term value of 93 is most close to: A. 13 B. 23 C. 36 D. 51



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PROBLEM 7: The third term of an arithmetic sequence with a 4"# term value of 65 and an 8"# term value of 93 is most close to: A. 15 B. 25 C. 58 D. 66



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PROBLEM 8: The first term of an arithmetic sequence with a 4"# term value of 65 and an 8"# term value of 93 is most close to: A. 44 B. 48 C. 56 D. 63



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PROBLEM 9: Determine the 26"# term of an arithmetic sequence given: 𝑎1 = 5 𝑎34 = 159 A. 7 B. 500 C. 467 D. 5



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PROBLEM 10: If the first term of an arithmetic sequence is equal to 6 and the common difference is equal to 3, the formula for the 𝑛"# term is best represented as: A. 6 + 153 𝑛 − 1 B. 6 + 3 𝑛 − 1 C. 6 − 3 𝑛 − 1 D. 153 + 3 𝑛 − 1



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ARITHMETIC PROGRESSION | SOLUTIONS SOLUTION 1: The GENERAL FORMULA for the 𝑛"# TERM of an ARITHMETIC 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 𝑛"# term of an arithmetic progression, is represented by the expression: 𝑙 =𝑎+ 𝑛−1 𝑑 Although not the most favorable way to present this formula, it’s what we have to work with, so let’s get to defining what each variable represents: • l: The 𝑛"# term, more commonly written as 𝑎8 • a: The first term in the sequence, more commonly written as 𝑎3 • n: The number of terms • d: The common difference of the sequence Knowing the definition of each variable, let’s slightly tweak the formula to read: 𝑎8 = 𝑎3 + 𝑛 − 1 𝑑 That’s much better, and probably something we are all more accustomed to seeing.

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Knowing that we are dealing with an ARITHMETIC PROGRESSION and that the terms 𝑎> and 𝑎? are four terms apart in this sequence, we can conclude that: 𝑎? = 𝑎> + 4𝑑 The problem defines: 𝑎> = 65 𝑎? = 93 This allows us to determine the COMMON DIFFERENCE, such that: 93 = 65 + 4𝑑 Rearranging to isolate and solve for the COMMON DIFFERENCE “𝑑”, we determine that: 𝑑=7 Revisiting the general formula for the nth term of this ARITHMETIC PROGRESSION, we can plug this value in for d, such that: 𝑎8 = 𝑎3 + 7 𝑛 − 1 Let’s hone in on determining what our first term, 𝑎3 , is.



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Using what has been established as the 4"# term, we can formulate the equation: 𝑎> = 𝑎3 + 7 𝑛 − 1 Which can be written as: 65 = 𝑎3 + 7 4 − 1 Rearranging to isolate and solve for 𝑎3 we find that: 𝑎3 = 44 Highlighting the data we have up to this point: 𝑎3 = 44 𝑎> = 65 𝑎? = 93 𝑑=7 We can conclude that the 𝑛"# term of this sequence can be represented as: 𝑎8 = 44 + 7 𝑛 − 1 The correct answer choice is C. 𝟒𝟒 + 𝟕 𝒏 − 𝟏



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SOLUTION 2: The GENERAL FORMULA for the 𝑛"# TERM of an ARITHMETIC 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 𝑛"# term of an arithmetic progression, is represented by the expression: 𝑙 =𝑎+ 𝑛−1 𝑑 Although not the most favorable way to present this formula, it’s what we have to work with, so let’s get to defining what each variable represents: • l: The 𝑛"# term, more commonly written as 𝑎8 • a: The first term in the sequence, more commonly written as 𝑎3 • n: The number of terms • d: The common difference of the sequence Knowing the definition of each variable, let’s slightly tweak the formula to read: 𝑎8 = 𝑎3 + 𝑛 − 1 𝑑 That’s much better, and probably something we are all more accustomed to seeing.



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Knowing that we are dealing with an ARITHMETIC PROGRESSION and that the terms 𝑎1 and 𝑎34 are seven terms apart in this sequence, we can conclude that: 𝑎34 = 𝑎1 + 7𝑑 The problem defines: 𝑎1 = 5 𝑎34 = 159 This allows us to determine the COMMON DIFFERENCE, such that: 159 = 5 + 7𝑑 Rearranging to isolate and solve for the COMMON DIFFERENCE “𝑑”, we determine that: 𝑑 = 22 Revisiting the general formula for the nth term of this ARITHMETIC PROGRESSION, we can plug this value in for d, such that: 𝑎8 = 𝑎3 + 22 𝑛 − 1 Let’s hone in on determining what our first term, 𝑎3 , is.



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Using what has been established as the 5"# term, we can formulate the equation: 𝑎1 = 𝑎3 + 22 𝑛 − 1 Which can be written as: 5 = 𝑎3 + 22 5 − 1 Rearranging to isolate and solve for 𝑎3 we find that: 𝑎3 = −83 Highlighting the data we have up to this point: 𝑎3 = −83 𝑎1 = 5 𝑎34 = 159 𝑑 = 22 We can conclude that the 𝑛"# term of this sequence can be represented as: 𝑎8 = −83 + 22 𝑛 − 1 The correct answer choice is A. −𝟖𝟑 + 𝟐𝟐 𝒏 − 𝟏



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SOLUTION 3: The FORMULA for the 𝑛"# TERM of an ARITHMETIC 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 𝑛"# term of an arithmetic progression, is represented by the expression: 𝑙 =𝑎+ 𝑛−1 𝑑 Although not the most favorable way to present this formula, it’s what we have to work with, so let’s get to defining what each variable represents: • l: The 𝑛"# term, more commonly written as 𝑎8 • a: The first term in the sequence, more commonly written as 𝑎3 • n: The number of terms • d: The common difference of the sequence Knowing the definition of each variable, let’s slightly tweak the formula to read: 𝑎8 = 𝑎3 + 𝑛 − 1 𝑑 That’s much better, and probably something we are all more accustomed to seeing.



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The first four terms of the sequence are: 𝑎3 = .25 1 + 2 = 2.25 𝑎4 = .25 2 + 2 = 2.50 𝑎G = .25 3 + 2 = 2.75 𝑎> = .25 4 + 2 = 3.00 From routine observation we note that the COMMON DIFFERENCE “𝑑” is: 𝑑 = .25 The GENERAL FORMULA for the SUM of the 𝑛"# TERM of an ARITHMETIC 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 sum of 𝑛"# term of an arithmetic progression, is represented by the expression:

𝑆=

𝑛(𝑎 + 𝑙) 𝑛[2𝑎 + 𝑛 − 1 𝑑] = 2 2

Where: 1. The first term is “𝑎”. 2. The common difference is “𝑑”. 3. The number of terms is “𝑛”. 4. The last or 𝑛"# term is “𝑙”. 5. The sum of 𝑛 terms is “𝑆”.

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This can be simplified in to more familiar terms as:

𝑆8 =

𝑛 (𝑎 + 𝑎8 ) 2 3

Up to this point we have defined: 𝑠8 = 21 𝑎3 = 2.25 𝑎8 = .25𝑛 + 2 Plugging these values into the general formula, we have:

21 =

𝑛 2.25 + . 25𝑛 + 2 2

Expanding a bit: 42 = 𝑛(2.25 + .25𝑛 + 2) Which gives us the QUADRATIC: . 25𝑛4 + 4.25𝑛 − 42 = 0 Solving the quadratic we find that: 𝑛 = −24 𝑛=7

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We know that a negative number can’t be the value we are after, which allows us to conclude that 𝑛 = 7. The correct answer choice is A. 𝟕

SOLUTION 4: The GENERAL FORMULA for the 𝑛"# TERM of an ARITHMETIC 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 𝑛"# term of an arithmetic progression, is represented by the expression: 𝑙 =𝑎+ 𝑛−1 𝑑 Although not the most favorable way to present this formula, it’s what we have to work with, so let’s get to defining what each variable represents: • l: The 𝑛"# term, more commonly written as 𝑎8 • a: The first term in the sequence, more commonly written as 𝑎3 • n: The number of terms • d: The common difference of the sequence



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Knowing the definition of each variable, let’s slightly tweak the formula to read: 𝑎8 = 𝑎3 + 𝑛 − 1 𝑑 That’s much better, and probably something we are all more accustomed to seeing. The problem gives us: 𝑎3 = 6 𝑑=3 This allows us to rewrite the general formula for the nth term of this ARITHMETIC PROGRESSION as: 𝑎8 = 6 + 3 𝑛 − 1 Calculating the 50"# term, we get: 𝑎1N = 6 + 3 50 − 1 = 153 The correct answer choice is D. 𝟏𝟓𝟑



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SOLUTION 5: The GENERAL FORMULA for the 𝑛"# TERM of an ARITHMETIC 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 𝑛"# term of an arithmetic progression, is represented by the expression: 𝑙 =𝑎+ 𝑛−1 𝑑 Although not the most favorable way to present this formula, it’s what we have to work with, so let’s get to defining what each variable represents: • l: The 𝑛"# term, more commonly written as 𝑎8 • a: The first term in the sequence, more commonly written as 𝑎3 • n: The number of terms • d: The common difference of the sequence Knowing the definition of each variable, let’s slightly tweak the formula to read: 𝑎8 = 𝑎3 + 𝑛 − 1 𝑑 That’s much better, and probably something we are all more accustomed to seeing.



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The first few terms of a sequence of positive integers divisible by 5 is given by: 𝑎3 = 5 𝑎4 = 10 𝑎G = 15 𝑎> = 20 By observation, we can conclude that: 𝑎3 = 5 𝑑=5 This allows us to rewrite the general formula for the nth term of this ARITHMETIC PROGRESSION as: 𝑎8 = 5 + 5 𝑛 − 1 We need to know how many terms in to this sequence the number 1,555 falls. To do this, we can set up the formula: 1,555 = 5 + 5 𝑛 − 1 Solving for “𝑛”, we determine that: 𝑛 = 311



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The GENERAL FORMULA for the SUM of the 𝑛"# TERM of an ARITHMETIC 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 sum up to the 𝑛"# term of an arithmetic progression, is represented by the expression:

𝑆=

𝑛(𝑎 + 𝑙) 𝑛[2𝑎 + 𝑛 − 1 𝑑] = 2 2

By now, we know that we can simplify this formula to read as:

𝑆8 =

𝑛 (𝑎 + 𝑎8 ) 2 3

Up to this point we have defined: 𝑛 = 311 𝑎3 = 5 𝑎8 = 1,555 Plugging this data in to our general formula, we get:

𝑆G33 =

311 (5 + 1555) = 242,580 2

The correct answer choice is B. 𝟐𝟒𝟐, 𝟓𝟖𝟎



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SOLUTION 6: The GENERAL FORMULA for the 𝑛"# TERM of an ARITHMETIC 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 𝑛"# term of an arithmetic progression, is represented by the expression: 𝑙 =𝑎+ 𝑛−1 𝑑 Although not the most favorable way to present this formula, it’s what we have to work with, so let’s get to defining what each variable represents: • l: The 𝑛"# term, more commonly written as 𝑎8 • a: The first term in the sequence, more commonly written as 𝑎3 • n: The number of terms • d: The common difference of the sequence Knowing the definition of each variable, let’s slightly tweak the formula to read: 𝑎8 = 𝑎3 + 𝑛 − 1 𝑑 That’s much better, and probably something we are all more accustomed to seeing.



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Knowing that we are dealing with an ARITHMETIC PROGRESSION and that the terms 𝑎> and 𝑎? are four terms apart in this sequence, we can conclude that: 𝑎? = 𝑎> + 8 − 4 𝑑 The problem defines: 𝑎> = 65 𝑎? = 93 This allows us to determine the COMMON DIFFERENCE, such that: 93 = 65 + 4𝑑 Rearranging to isolate and solve for the COMMON DIFFERENCE “𝑑”, we determine that: 𝑑=7 Revisiting the general formula for the nth term of this ARITHMETIC PROGRESSION, we can plug this value in for d, such that: 𝑎8 = 𝑎3 + 7 𝑛 − 1 Let’s hone in on determining what our first term, 𝑎3 , is.



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Using what has been established as the 4"# term, we can formulate the equation: 𝑎> = 𝑎3 + 7 𝑛 − 1 Which can be written as: 65 = 𝑎3 + 7 4 − 1 Rearranging to isolate and solve for 𝑎3 we find that: 𝑎3 = 44 Highlighting the data we have up to this point: 𝑎3 = 44 𝑎> = 65 𝑎? = 93 𝑑=7 Revisiting the general formula for the nth term of this ARITHMETIC PROGRESSION, we can plug this value in for d, such that: 𝑎8 = 𝑎3 + 7 𝑛 − 1 We are asked to determine what the second term of this progression would be.



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This is simple plug and play at this point, giving us: 𝑎4 = 44 + 7 2 − 1 = 51 The correct answer choice is D. 𝟓𝟏

SOLUTION 7: The GENERAL FORMULA for the 𝑛"# TERM of an ARITHMETIC 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 𝑛"# term of an arithmetic progression, is represented by the expression: 𝑙 =𝑎+ 𝑛−1 𝑑 Although not the most favorable way to present this formula, it’s what we have to work with, so let’s get to defining what each variable represents: • l: The 𝑛"# term, more commonly written as 𝑎8 • a: The first term in the sequence, more commonly written as 𝑎3 • n: The number of terms • d: The common difference of the sequence



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Knowing the definition of each variable, let’s slightly tweak the formula to read: 𝑎8 = 𝑎3 + 𝑛 − 1 𝑑 That’s much better, and probably something we are all more accustomed to seeing. Knowing that we are dealing with an ARITHMETIC PROGRESSION and that the terms 𝑎> and 𝑎? are four terms apart in this sequence, we can conclude that: 𝑎? = 𝑎> + 8 − 4 𝑑 The problem defines: 𝑎> = 65 𝑎? = 93 This allows us to determine the COMMON DIFFERENCE, such that: 93 = 65 + 4𝑑 Rearranging to isolate and solve for the COMMON DIFFERENCE “𝑑”, we determine that: 𝑑=7



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Revisiting the general formula for the nth term of this ARITHMETIC PROGRESSION, we can plug this value in for d, such that: 𝑎8 = 𝑎3 + 7 𝑛 − 1 Let’s hone in on determining what our first term, 𝑎3 , is. Using what has been established as the 4"# term, we can formulate the equation: 𝑎> = 𝑎3 + 7 𝑛 − 1 Which can be written as: 65 = 𝑎3 + 7 4 − 1 Rearranging to isolate and solve for 𝑎3 we find that: 𝑎3 = 44 Highlighting the data we have up to this point: 𝑎3 = 44 𝑎> = 65 𝑎? = 93 𝑑 = 9.33



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We can conclude that the 𝑛"# term of this sequence can be represented as: 𝑎8 = 44 + 7 𝑛 − 1 We are asked to determine what the third term of this progression would be. This is simple plug and play at this point, giving us: 𝑎G = 44 + 7 3 − 1 = 58 The correct answer choice is C. 𝟓𝟖

SOLUTION 8: The GENERAL FORMULA for the 𝑛"# TERM of an ARITHMETIC 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 𝑛"# term of an arithmetic progression, is represented by the expression: 𝑙 =𝑎+ 𝑛−1 𝑑



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Although not the most favorable way to present this formula, it’s what we have to work with, so let’s get to defining what each variable represents: • l: The 𝑛"# term, more commonly written as 𝑎8 • a: The first term in the sequence, more commonly written as 𝑎3 • n: The number of terms • d: The common difference of the sequence Knowing the definition of each variable, let’s slightly tweak the formula to read: 𝑎8 = 𝑎3 + 𝑛 − 1 𝑑 That’s much better, and probably something we are all more accustomed to seeing. Knowing that we are dealing with an ARITHMETIC PROGRESSION and that the terms 𝑎> and 𝑎? are four terms apart in this sequence, we can conclude that: 𝑎? = 𝑎> + 4𝑑 The problem defines: 𝑎> = 65 𝑎? = 93 This allows us to determine the COMMON DIFFERENCE, such that: 93 = 65 + 4𝑑

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Rearranging to isolate and solve for the COMMON DIFFERENCE “𝑑”, we determine that: 𝑑=7 Revisiting the general formula for the nth term of this ARITHMETIC PROGRESSION, we can plug this value in for d, such that: 𝑎8 = 𝑎3 + 7 𝑛 − 1 Let’s hone in on determining what our first term, 𝑎3 , is. Using what has been established as the 4"# term, we can formulate the equation: 𝑎> = 𝑎3 + 7 𝑛 − 1 Which can be written as: 65 = 𝑎3 + 7 4 − 1 Rearranging to isolate and solve for 𝑎3 we find that: 𝑎3 = 44 The correct answer choice is A. 𝟒𝟒



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SOLUTION 9: The GENERAL FORMULA for the 𝑛"# TERM of an ARITHMETIC 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 𝑛"# term of an arithmetic progression, is represented by the expression: 𝑙 =𝑎+ 𝑛−1 𝑑 Although not the most favorable way to present this formula, it’s what we have to work with, so let’s get to defining what each variable represents: • l: The 𝑛"# term, more commonly written as 𝑎8 • a: The first term in the sequence, more commonly written as 𝑎3 • n: The number of terms • d: The common difference of the sequence Knowing the definition of each variable, let’s slightly tweak the formula to read: 𝑎8 = 𝑎3 + 𝑛 − 1 𝑑 That’s much better, and probably something we are all more accustomed to seeing. Knowing that we are dealing with an ARITHMETIC PROGRESSION and that the terms 𝑎1 and 𝑎34 are seven terms apart in this sequence, we can conclude that: Made with by Prepineer | Prepineer.com

𝑎34 = 𝑎1 + 7𝑑 The problem defines: 𝑎1 = 5 𝑎34 = 159 This allows us to determine the COMMON DIFFERENCE, such that: 159 = 5 + 7𝑑 Rearranging to isolate and solve for the COMMON DIFFERENCE “𝑑”, we determine that: 𝑑 = 22 Revisiting the general formula for the nth term of this ARITHMETIC PROGRESSION, we can plug this value in for d, such that: 𝑎8 = 𝑎3 + 22 𝑛 − 1 Let’s hone in on determining what our first term, 𝑎3 , is. Using what has been established as the 5"# term, we can formulate the equation: 𝑎1 = 𝑎3 + 22 𝑛 − 1

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Which can be written as: 5 = 𝑎3 + 22 5 − 1 Rearranging to isolate and solve for 𝑎3 we find that: 𝑎3 = −83 Highlighting the data we have up to this point: 𝑎3 = −83 𝑎1 = 5 𝑎34 = 159 𝑑 = 22 We can conclude that the 𝑛"# term of this sequence can be represented as: 𝑎8 = −83 + 22 𝑛 − 1 The 26th term can then be calculated as: 𝑎4Q = −83 + 22 26 − 1 = 467 The correct answer choice is C. 𝟒𝟔𝟕



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SOLUTION 10: The GENERAL FORMULA for the 𝑛"# TERM of an ARITHMETIC 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 𝑛"# term of an arithmetic progression, is represented by the expression: 𝑙 =𝑎+ 𝑛−1 𝑑 Although not the most favorable way to present this formula, it’s what we have to work with, so let’s get to defining what each variable represents: • l: The 𝑛"# term, more commonly written as 𝑎8 • a: The first term in the sequence, more commonly written as 𝑎3 • n: The number of terms • d: The common difference of the sequence Knowing the definition of each variable, let’s slightly tweak the formula to read: 𝑎8 = 𝑎3 + 𝑛 − 1 𝑑 That’s much better, and probably something we are all more accustomed to seeing.



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The problem gives us: 𝑎3 = 6d 𝑑=3 This allows us to rewrite the general formula for the nth term of this ARITHMETIC PROGRESSION as: 𝑎8 = 6 + 3 𝑛 − 1 The correct answer choice is B. 𝟔 + 𝟑 𝒏 − 𝟏



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