54.1 Arithmetic Progression Concept Overview

ARITHMETIC PROGRESSION | CONCEPT OVERVIEW The topic of ARITHMETIC PROGRESSION can be referenced on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

CONCEPT INTRO: A SEQUENCE is an ORDERED LIST of NUMBERS that can be put in the form 𝑎! , 𝑎! , 𝑎! , 𝑎! , … , 𝑎! . The numbers represented in the sequence are called TERMS or ELEMENTS. A SERIES is the value obtained when you ADD UP all the TERMS of a sequence. A series is represented as a SUMMATION to represent the sum of the terms contained in the sequence, such that: !

𝑎! !!!

Where: • The LOWER INDEX is represented by 𝑛 = 0 and indicates the counter value of 𝑛 that we should start counting from. • The UPPER INDEX, denoted as 𝑛, tells us the last counter of n, which represents the last term of the summation as 𝑎! .

Made with

by Prepineer | Prepineer.com

• 𝑛 is the COUNTING VARIABLE used to represent the indices of the progression. It is common to use the variables 𝑖, 𝑗, 𝑘, and 𝑛 to represent these indices. An example of something we may encounter on the exam is a series written in SUMMATION FORM, such as: !

𝑎! !!!

This is read as the sum of the progression from n equals 1 to n equals 5 of the progression represented by the summation expression, 𝑎! . Writing out all of the terms in the EXPANDED FORM of the progression, we find the terms lay out as: !

𝑎! = 𝑎! + 𝑎! + 𝑎! + 𝑎! + 𝑎! !!!

An ARITHMETIC PROGRESSION is a progression of a sequence where each consecutive term is derived from the previous term by addition or subtraction of a fixed number called a COMMON DIFFERENCE. The PATTERN of such a sequence is that we are always ADDING, or SUBTRACTING, a FIXED NUMBER to the PREVIOUS TERM to get to the next term.



Made with

by Prepineer | Prepineer.com

Every progression, or sequence, that has a pattern in addition isn’t necessarily an arithmetic sequence; it only is if you are always adding the SAME number each time. An example of an ARITHMETIC PROGRESSION in a given series is written in expanded form as: 𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, … , 𝑎 + 𝑛 − 1 𝑑 To determine whether a given finite sequence of numbers is an arithmetic progression, subtract each number from the number that precedes it. If the differences are uniform and equal across all terms, the series is arithmetic. 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. In this formula, the variables are defined as: • a: The first term in the sequence • d: The common difference of the sequence

Made with

by Prepineer | Prepineer.com

• n: The number of terms • l: The 𝑛!! term The GENERAL FORMULA for the SUM up to 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 all terms 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

Again, not the most favorable way to present this formula, but working along with it, the variables are defined as: • a: The first term in the sequence • d: The common difference of the sequence • n: The number of terms • l: The 𝑛!! term 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.



Made with

by Prepineer | Prepineer.com

The FIVE ELEMENTS of an ARITHMETIC PROGRESSION and the letters by which they are denoted are: 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”. The GENERAL FORMULA for the ARITHMETIC MEAN is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook. The ARITHMETIC MEAN, or the AVERAGE, between two numbers is the middle term of an arithmetic progression. The arithmetic mean is defined as the sum of the numbers divided by the quantity of terms in the series. The ARITHMETIC MEAN of an arithmetic progression is represented by the expression:

𝑆=



𝑛 (𝑎 + 𝑙) 2

Made with

by Prepineer | Prepineer.com

ARITHMETIC PROGRESSION | SOLUTIONS The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material. The value of the following summation is most close to: !"

2𝑛 − 5 !!!"

A. 1881 B. 8118 C. 105 D. 4442

SOLUTION: The TOPIC of ARITHMETIC PROGRESSIONS can be referenced under the subject of MATHEMATICS on page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We are given the expression: !"

2𝑛 − 5 !!!"



Made with

by Prepineer | Prepineer.com

And are asked to determine the result starting at the 15th term and up to through the 47th term. Many will proceed in this problem by deriving the values for each of the terms between 15 and 47 and adding them all up. This is an option, but by far, the most time consuming of all. We know that we are dealing with an ARITHMETIC PROGRESSION here. An ARITHMETIC PROGRESSION is a progression of a sequence where each consecutive term is derived from the previous term by addition or subtraction of a fixed number called a COMMON DIFFERENCE. The PATTERN of such a sequence is that we are always ADDING, or SUBTRACTING, a FIXED NUMBER to the PREVIOUS TERM to get to the next term. Flipping over to Page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing, we are given the GENERAL FORMULA for the SUM up to the 𝑛!! TERM of an ARITHMETIC PROGRESSION as:

𝑆=



𝑛(𝑎 + 𝑙) 𝑛[2𝑎 + 𝑛 − 1 𝑑] = 2 2

Made with

by Prepineer | Prepineer.com

For the sake of creating a formula that is more user friendly, let’s rewrite what we are given as:

𝑆! =

𝑛 (𝑎 + 𝑎! ) 2 !

So the strategy will be this, instead of adding all the terms up between 15 and 47, we will determine the SUM up through the 47th value as well as the SUM up through the 14th value. We will then subtract the SUM up through the 14th value from the SUM up through the 47th value to get our final answer. The formula given to us in the NCEES Reference Handbook requires that we get a few terms defined, specifically, the 1st term, the 14th term, and the 47th term. 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 formula: 𝑙 =𝑎+ 𝑛−1 𝑑 Here the variable “a” represents the first term, so let’s derive that.



Made with

by Prepineer | Prepineer.com

Simply taking the original sequence we are given, we plug in the value for n=1 and we get: 𝑎! = 2 1 − 5 = −3 Now to get the value of the 14th term, we need to determine the COMMON DIFFERENCE, d, of the ARITHMETIC PROGRESSION. We can determine this by establishing the next few terms in the SEQUENCE, and observing what this difference is. Doing so we get: 𝑎! = 2 2 − 5 = −1 𝑎! = 2 3 − 5 = 1 𝑎! = 2 4 + 5 = 3 From observation, we can see that each term will be two units larger than the previous, so this is an ARITHMETICAL SEQUENCE where the COMMON DIFFERENCE 𝑑 = 2. We now have: 𝑎! = −3 𝑑=2



Made with

by Prepineer | Prepineer.com

We need the 14th term and we need the 47th term…let’s start with the 14th. Plugging in our values we get: 𝑎!" = 𝑙 = −3 + 14 − 1 2 = 23 Now defining the 47th term, we get: 𝑎!" = 𝑙 = −3 + 47 − 1 2 = 89 Recapping the data we have defined, we have: 𝑎! = −3 𝑑=2 𝑎!" = 23 𝑎!" = 89 We now have to SUM all the values up to both through the 14th term as well as the 47th term. We can flip to page 30 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing and reference the GENERAL FORMULA for the SUM of the 𝑛!! TERM of an ARITHMETIC PROGRESSION, which is generally stated as:

𝑆=



𝑛(𝑎 + 𝑙) 𝑛[2𝑎 + 𝑛 − 1 𝑑] = 2 2

Made with

by Prepineer | Prepineer.com

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” This can be simplified as:

𝑆! =

𝑛 (𝑎 + 𝑎! ) 2 !

We have all the data we need up to this point, so let’s move forward with our calculations. Summing up to through the 14th term we have:

𝑆!" =

14 −3 + 23 = 140 2

Summing up to through the 47th term we have:

𝑆!" =



47 −3 + 89 = 2021 2

Made with

by Prepineer | Prepineer.com

Now we don’t want the SUM of all 47 terms, but only those between 15 and 47…so we subtracting the SUM up through the 14th term with the SUM up through the 47th term such that: !"

2𝑛 − 5 = 2021 − 140 = 1881 !!!"

Therefore: !"

2𝑛 − 5 = 1881 !!!

The correct answer choice is A. 𝟏𝟖𝟖𝟏



Made with

by Prepineer | Prepineer.com