Which Sequence Represents An Arithmetic Sequence? (Question)

An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5,7,9,11,13,⋯ 5, 7, 9, 11, 13, ⋯ is an arithmetic sequence with common difference of 2.

Contents

What are the 5 examples of arithmetic sequence?

= 3, 6, 9, 12,15,. A few more examples of an arithmetic sequence are: 5, 8, 11, 14, 80, 75, 70, 65, 60,

What are 2 examples of arithmetic sequence?

An arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term. For example in the arithmetic sequence 3, 9, 15, 21, 27, the common difference is 6.

How do you find the arithmetic sequence?

The arithmetic sequence formula is given as, an=a1+(n−1)d a n = a 1 + ( n − 1 ) d where, an a n = a general term, a1 a 1 = first term, and and d is the common difference. This is to find the general term in the sequence.

What is the common difference arithmetic sequence?

In arithmetic sequences, the common difference is simply the value that is added to each term to produce the next term of the sequence.

What is arithmetic sequence and series?

An arithmetic sequence is a sequence where the difference d between successive terms is constant. The general term of an arithmetic sequence can be written in terms of its first term a1, common difference d, and index n as follows: an=a1+(n−1)d. An arithmetic series is the sum of the terms of an arithmetic sequence.

What sequence is formed by the reciprocals of the arithmetic sequence?

A harmonic progression is a sequence of real numbers formed by taking the reciprocals of an arithmetic progression. Equivalently, it is a sequence of real numbers such that any term in the sequence is the harmonic mean of its two neighbors.

Which of the following is not arithmetic sequence?

The following are not examples of arithmetic sequences: 1.) 2,4,8,16 is not because the difference between first and second term is 2, but the difference between second and third term is 4, and the difference between third and fourth term is 8. No common difference so it is not an arithmetic sequence.

How can I differentiate an arithmetic sequence from other forms?

An arithmetic sequence is a sequence of numbers that is calculated by subtracting or adding a fixed term to/from the previous term. However, a geometric sequence is a sequence of numbers where each new number is calculated by multiplying the previous number by a fixed and non-zero number.

Introduction to Arithmetic Progressions

Generally speaking, a progression is a sequence or series of numbers in which the numbers are organized in a certain order so that the relationship between the succeeding terms of a series or sequence remains constant. It is feasible to find the n thterm of a series by following a set of steps. There are three different types of progressions in mathematics:

  1. Arithmetic Progression (AP), Geometric Progression (GP), and Harmonic Progression (HP) are all types of progression.

AP, also known as Arithmetic Sequence, is a sequence or series of integers in which the common difference between two subsequent numbers in the series is always the same. As an illustration: Series 1: 1, 3, 5, 7, 9, and 11. Every pair of successive integers in this sequence has a common difference of 2, which is always true. Series 2: numbers 28, 25, 22, 19, 16, 13,. There is no common difference between any two successive numbers in this series; instead, there is a strict -3 between any two consecutive numbers.

Terminology and Representation

  • Common difference, d = a 2– a 1= a 3– a 2=. = a n– a n – 1
  • A n= n thterm of Arithmetic Progression
  • S n= Sum of first n elements in the series
  • A n= n

General Form of an AP

Given that ais treated as a first term anddis treated as a common difference, the N thterm of the AP may be calculated using the following formula: As a result, using the above-mentioned procedure to compute the n terms of an AP, the general form of the AP is as follows: Example: The 35th term in the sequence 5, 11, 17, 23,. is to be found. Solution: When looking at the given series,a = 5, d = a 2– a 1= 11 – 5 = 6, and n = 35 are the values. As a result, we must use the following equations to figure out the 35th term: n= a + (n – 1)da n= 5 + (35 – 1) x 6a n= 5 + 34 x 6a n= 209 As a result, the number 209 represents the 35th term.

Sum of n Terms of Arithmetic Progression

The arithmetic progression sum is calculated using the formula S n= (n/2)

Derivation of the Formula

Arithmetic progression sum S n=(n/2) is calculated using the following formula:

Sample Problems on Arithmetic Progressions

Problem 1: Calculate the sum of the first 35 terms in the sequence 5,11,17,23, and so on. a = 5 in the given series, d = a 2–a in the provided series, and so on. The number 1 equals 11 – 5 = 6, and the number n equals 35. S n= (n/2)(2a + (n – 1) x d)(n/2)(2a + (n – 1) x d) S n= (35/2)(2 x 5 + (35 – 1) x 6)(35/2)(2 x 5 + (35 – 1) x 6) S n= (35/2)(10 + 34 x 6) n= (35/2)(10 + 34 x 6) n= (35/2)(10 + 34 x 6) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) A = 35214/2A = 3745S n= 35214/2A = 3745 Find the sum of a series where the first term of the series is 5 and the last term of the series is 209, and the number of terms in the series is 35, as shown in Problem 2.

Problem 2.

S n= (35/2)(5 + 209) S n= (35/2)(5 + 209) S n= (35/2)(5 + 209) A = 35214/2A = 3745S n= 35214/2A = 3745 Problem 3: A amount of 21 rupees is divided among three brothers, with each of the three pieces of money being in the AP and the sum of their squares being the sum of their squares being 155.

Solution: Assume that the three components of money are (a-d), a, and (a+d), and that the total amount allocated is in AP.

155 divided by two equals 155 Taking the value of ‘a’ into consideration, we obtain 3(7) 2+ 2d.

2= 4d = 2 = 2 The three portions of the money that was dispersed are as follows:a + d = 7 + 2 = 9a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5 As a result, the most significant portion is Rupees 9 million.

Arithmetic Sequences – Precalculus

Sequences, probability, and counting theory are all topics covered in this course.

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Learning Objectives

You will learn the following things in this section:

  • Calculate the common difference between two arithmetic sequences
  • Make a list of the terms in an arithmetic sequence
  • When dealing with an arithmetic series, use a recursive formula. When dealing with an arithmetic series, use an explicit formula.

Large purchases, such as computers and automobiles, are frequently made by businesses for their own use. For taxation reasons, the book-value of these supplies diminishes with each passing year. Depreciation is the term used to describe this decline in value. Depreciation may be calculated in several ways, one of which is straight-line depreciation, which means that the value of the asset drops by the same amount each year. Consider the case of a lady who decides to start her own modest contracting firm.

She expects to be able to sell the truck for $8,000 after five years, according to her estimations.

After one year, the vehicle will be worth?21,600; after two years, it will be worth?18,200; after three years, it will be worth?14,800; after four years, it will be worth?11,400; and after five years, it will be worth?8,000.

Finding Common Differences

The values of the vehicle in the example are said to constitute an anarithmetic sequence since they vary by a consistent amount each year, according to the definition. Every term grows or decreases by the same constant amount, which is referred to as the common difference of the sequence. –3,400 is the common difference between the two sequences in this case. Another example of an arithmetic series may be seen in the sequence below. In this situation, the constant difference is three times more than one.

Sequence of Arithmetic Operations When two successive words are added together, the difference between them is a constant.

Given that and is the initial term of an arithmetic sequence, and is the common difference, then the sequence will be as follows: Identifying Commonalities and Dissimilarities Is each of the sequences mathematical in nature?

To establish whether or not there is a common difference between two terms, subtract each phrase from the succeeding term.

  1. The series is not arithmetic because there is no common difference between the elements
  2. The sequence is arithmetic because there is a common difference between the elements. The most often encountered difference is 4

Analysis Each of these sequences is represented by a graph, which is depicted in (Figure). We can observe from the graphs that, despite the fact that both sequences exhibit increase, is not linear whereas is linear, as we previously said. Given that arithmetic sequences have an invariant rate of change, their graphs will always consist of points on a straight line. If we are informed that a series is arithmetic, do we have to subtract every term from the term after it in order to identify the common difference between the terms?

As long as we know that the sequence is arithmetic, we may take any one term from it and subtract it from the following term to determine the common difference.

If this is the case, identify the common difference. The sequence follows a mathematical pattern. The main distinction is whether or not the provided sequence is arithmetic. If this is the case, identify the common difference. Because of this, the sequence is not arithmetic.

Writing Terms of Arithmetic Sequences

After recognizing an arithmetic sequence, we can determine the terms if we are provided the first term as well as the common difference between the two terms. The terms may be discovered by starting with the first term and repeatedly adding the common difference to the end of the list. In addition, any term may be obtained by inserting the values of and into the formula below, which is shown below. Find the first many terms of an arithmetic series based on the first term and the common difference of the sequence.

  1. To determine the second term, add the common difference to the first term
  2. And so on. To determine the third term, add the common difference to the second term
  3. This will give you the third term. Make sure to keep going until you’ve found all of the needed keywords
  4. Create a list of words separated by commas and enclosed inside brackets

Creating Arithmetic Sequences in the Form of Terms Fill in the blanks with the first five terms of the arithmetic sequencewithand. Adding three equals the same as deleting three. Starting with the first phrase, remove 3 from each word to arrive at the next term in the sequence. The first five terms are as follows: Analysis As predicted, the graph of the series is made up of points on a line, as seen in the figure (Figure). List the first five terms of the arithmetic sequence beginning with and ending with and.

There are two ways to write the sequence: in terms of the first word, 8, and the common difference.

We can identify the commonalities between the two situations.

Analysis Observe how each term’s common difference is multiplied by one in order to identify the following terms: once to find the second term, twice to get the third term, and so on.

Using Recursive Formulas for Arithmetic Sequences

With the use of a recursive formula, several arithmetic sequences may be defined in terms of the preceding term. The formula contains an algebraic procedure that may be used to determine the terms of the series. We can discover the next term in an arithmetic sequence by utilizing a function of the term that came before it using a recursive formula. In each term, the previous term is multiplied by the common difference, and so on. Suppose the common difference is 5, and each phrase is the preceding term multiplied by five.

For an Arithmetic Sequence, a Recursive Formula can be used.

  1. To discover the common difference between two terms, subtract any phrase from the succeeding term. In the recursive formula for arithmetic sequences, start with the initial term and substitute the common difference

The common difference can be found by subtracting any phrase from the succeeding term. In the recursive formula for arithmetic sequences, start with the initial term and substitute the common difference.

Using Explicit Formulas for Arithmetic Sequences

It is possible to think of anarithmetic sequence as a function on the domain of natural numbers; it is a linear function since the rate of change remains constant throughout the series. The constant rate of change, often known as the slope of the function, is the most frequently seen difference. If we know the slope and the vertical intercept of a linear function, we can create the function. For the -intercept of the function, we may take the common difference from the first term in the sequence and remove it from the result.

The common difference is that the series indicates a linear function with a slope of, whereas the difference is that We subtract from in order to determine the-intercept.

The graph is displayed in the following: (Figure).

The following equation is obtained by substituting for the slope as well as for the vertical intercept: To create an explicit formula for an arithmetic series, we do not need to identify the vertical intercept of the sequence.

  1. Identify the commonalities and differences
  2. Replace the common difference and the first word with the following:

After that, I’ll write the term paper. An Arithmetic Sequence with a Clearly Defined Formula Create an explicit formula for the arithmetic series using the following syntax: It is possible to calculate the common difference by subtracting the first term from the second term. The most often encountered difference is ten. To simplify the formula, substitute the common difference and the first term in the series into it. Analysis It can be shown in (Figure) that the slope of this sequence is 10 and that the vertical intercept is 10.

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Finding the Number of Terms in a Finite Arithmetic Sequence

When determining the number of terms in a finite arithmetic sequence, explicit formulas can be employed to make the determination. Finding the common difference and determining the number of times the common difference must be added to the first term in order to produce the last term of the sequence are both necessary steps. The Number of Terms in a Finite Arithmetic Sequence can be determined by the following method: The number of terms in the infinite arithmetic sequence is to be determined.

The most noticeable change is.

substitute for and find a solution for There are a total of eight terms in the series.

There are a total of 11 terms included in the sequence.

Solving Application Problems with Arithmetic Sequences

Using an initial term ofinstead of in many application difficulties makes logical sense in many situations In order to account for the variation in beginning terms in both cases, we make a little modification to the explicit formula. The following is the formula that we use: Problem-Solving using Arithmetic Sequences in Practical Situations Week after week, a youngster of five years old receives an allowance of one dollar. His parents have promised him a?2 per week rise on a yearly basis.

  1. Create a method for calculating the child’s weekly stipend over the course of a year
  2. What will be the child’s allowance when he reaches the age of sixteen
  1. In this case, an arithmetic sequence with a starting term of 1 and a common difference of 2 may be used to represent what happened. Let be the amount of the allowance, and let be the number of years after reaching the age of five years. Using the modified explicit formula for an arithmetic series, we get the following results: By subtracting, we may find out how many years have passed since we were five. We’re asking for the child’s allowance after 11 years of being without one. In order to calculate the child’s allowance at the age of 16, substitute 11 into the calculation. What will the child’s allowance be when he or she reaches the age of sixteen? 23 hours each week

Using an arithmetic sequence starting with 1 and ending with 2 as the common difference, the situation may be represented mathematically. The amount of the allowance and the number of years beyond age 5 will be denoted by and, respectively, and. Using the modified explicit formula for an arithmetic series, we may obtain the following results: By subtracting, we may find out how many years have passed since age 5. After 11 years, we are asking for the child’s allowance. To calculate the child’s allowance at the age of 16, substitute 11 into the calculation.

Every week, there are 23 hours.

Section Exercises

What is an arithmetic sequence, and how does it work? A series in which each subsequent term grows (or lowers) by a constant value is known as a constant value sequence. What is the procedure for determining the common difference of an arithmetic sequence? What is the best way to tell if a sequence is arithmetic or not? If the difference between all successive words is the same, we may say that they are sequential. This is the same as saying that the sequence has a common difference between it and the rest of the series.

What are the primary distinctions between the two methods?

What is the difference between them?

These two types of functions are distinct because their domains are not the same; linear functions are defined for all real numbers, whereas arithmetic sequences are specified for natural numbers or a subset of the natural numbers, respectively.

Algebraic

Work on finding the common difference for the arithmetic sequence that is supplied in the following problems. The most noticeable distinction is Determine if the sequence in the following exercises is mathematical or not in the following exercises. If this is the case, identify the common difference. Because of this, the sequence is not arithmetic. Write the first five terms of the arithmetic sequence given the first term and the common difference in the following tasks. Fill in the blanks with the first five terms of the arithmetic series given two terms in the following problems.

  1. The first term is 3, the most common difference is 4, and the fifth term is 3.
  2. The first term is 5, the most common difference is 6, and the eighth term is The first term is 6, the most common difference is 7, and the sixth term is 6.
  3. To complete the following problems, you must discover the first term from an arithmetic series given two terms.
  4. Make use of the recursive formula in order to write the first five terms of the arithmetic sequence in each of the following tasks.
  5. The following activities require you to build a recursive formula for the provided arithmetic sequence, followed by a search for the required term in the formula.
  6. Find the fourteenth term.
  7. For the following problems, write the first five terms of the arithmetic sequence using the explicit formula that was provided.
  8. The number of terms in the finite arithmetic sequence presented in the following task is to be determined for the following exercises: The series has a total of ten terms.

Graphical

Determine whether or not the graph given reflects an arithmetic sequence in the following tasks. The graph does not reflect an arithmetic sequence in the traditional sense. To complete the following activities, use the information supplied to graph the first 5 terms of the arithmetic sequence using the information provided.

Technology

Determine whether or not the graph given reflects an arithmetic sequence in the following activities.

Although the graph appears to reflect an arithmetic sequence, it does not. As a result of this knowledge, graph the first five terms of the arithmetic sequence in the following exercise(s).

  • Select SEQ in the fourth line
  • Select DOT in the fifth line
  • Then press Enter or Return.
  • Press
  • Then press to go to theTBLSET
  • Then press to go to theTABLE
  • And so on.

In the column with the header “First Seven Terms,” what are the first seven terms listed? Use the scroll-down arrow to move to the next page. What is the monetary value assigned to Press. Set the parameters and then press. The sequence should be graphed exactly as it appears on the graphing calculator. Follow the techniques outlined above to work with the arithmetic sequence using a graphing calculator for the following problems. The first seven terms that appear in the column with the title in the TABLE feature are referred to as The sequence should be graphed exactly as it appears on the graphing calculator.

Extensions

Examples of arithmetic sequences with four terms that are the same are shown. Give two instances of arithmetic sequences whose tenth terms are as follows: There will be a range of responses. Examples: and The fifth term of the arithmetic series must be discovered Determine the eleventh term in the arithmetic series. At what point does the sequence reach the number 151? When does the series begin to contain negative values and at what point does it stop? For which terms does the finite arithmetic sequence have integer values?

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Create an arithmetic series using a recursive formula to demonstrate your understanding.

There will be a range of responses.

Example: Formula for recursion: The first four terms are as follows: Create an arithmetic sequence using an explicit formula to demonstrate your understanding.

Glossary

Arithmetic sequencea sequence in which the difference between any two consecutive terms is a constantcommon difference is a series in which the difference between any two consecutive terms is a constant an arithmetic series is the difference between any two consecutive words in the sequence

Arithmetic Sequences and Sums

A sequence is a collection of items (typically numbers) that are arranged in a specific order. Each number in the sequence is referred to as aterm (or “element” or “member” in certain cases); for additional information, see Sequences and Series.

Arithmetic Sequence

An Arithmetic Sequence is characterized by the fact that the difference between one term and the next is a constant. In other words, we just increase the value by the same amount each time. endlessly.

Example:

1, 4, 7, 10, 13, 16, 19, 22, and 25 are the numbers 1 through 25. Each number in this series has a three-digit gap between them. Each time the pattern is repeated, the last number is increased by three, as seen below: As a general rule, we could write an arithmetic series along the lines of

  • There are two words: Ais the first term, and dis is the difference between the two terms (sometimes known as the “common difference”).

Example: (continued)

1, 4, 7, 10, 13, 16, 19, 22, and 25 are the numbers 1 through 25. Has:

  • In this equation, A = 1 represents the first term, while d = 3 represents the “common difference” between terms.

And this is what we get:

Rule

It is possible to define an Arithmetic Sequence as a rule:x n= a + d(n1) (We use “n1” since it is not used in the first term of the sequence).

Example: Write a rule, and calculate the 9th term, for this Arithmetic Sequence:

3, 8, 13, 18, 23, 28, 33, and 38 are the numbers three, eight, thirteen, and eighteen. Each number in this sequence has a five-point gap between them. The values ofaanddare as follows:

  • A = 3 (the first term)
  • D = 5 (the “common difference”)
  • A = 3 (the first term).

Making use of the Arithmetic Sequencerule, we can see that_xn= a + d(n1)= 3 + 5(n1)= 3 + 3 + 5n 5 = 5n 2 xn= a + d(n1) = 3 + 3 + 3 + 5n n= 3 + 3 + 3 As a result, the ninth term is:x 9= 5 9 2= 43 Is that what you’re saying? Take a look for yourself! Arithmetic Sequences (also known as Arithmetic Progressions (A.P.’s)) are a type of arithmetic progression.

Advanced Topic: Summing an Arithmetic Series

To summarize the terms of this arithmetic sequence:a + (a+d) + (a+2d) + (a+3d) + (a+4d) + (a+5d) + (a+6d) + (a+7d) + (a+8d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + ( make use of the following formula: What exactly is that amusing symbol? It is referred to as The Sigma Notation is a type of notation that is used to represent a sigma function. Additionally, the starting and finishing values are displayed below and above it: “Sum upnwherengoes from 1 to 4,” the text states. 10 is the correct answer.

Example: Add up the first 10 terms of the arithmetic sequence:

The values ofa,dandnare as follows:

  • In this equation, A = 1 represents the first term, d = 3 represents the “common difference” between terms, and n = 10 represents the number of terms to add up.

As a result, the equation becomes:= 5(2+93) = 5(29) = 145 Check it out yourself: why don’t you sum up all of the phrases and see whether it comes out to 145?

Footnote: Why Does the Formula Work?

Let’s take a look at why the formula works because we’ll be employing an unusual “technique” that’s worth understanding. First, we’ll refer to the entire total as “S”: S = a + (a + d) +. + (a + (n2)d) +(a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n1)d) + (a + (n1)d) + After that, rewrite S in the opposite order: S = (a + (n1)d)+ (a + (n2)d)+. +(a + d)+a. +(a + d)+a. +(a + d)+a. Now, term by phrase, add these two together:

S = a + (a+d) + . + (a + (n-2)d) + (a + (n-1)d)
S = (a + (n-1)d) + (a + (n-2)d) + . + (a + d) + a
2S = (2a + (n-1)d) + (2a + (n-1)d) + . + (2a + (n-1)d) + (2a + (n-1)d)

Each and every term is the same! Furthermore, there are “n” of them. 2S = n (2a + (n1)d) = n (2a + (n1)d) Now, we can simply divide by two to obtain the following result: The function S = (n/2) (2a + (n1)d) is defined as This is the formula we’ve come up with:

Arithmetic Sequences

In mathematics, an arithmetic sequence is a succession of integers in which the value of each number grows or decreases by a fixed amount each term. When an arithmetic sequence has n terms, we may construct a formula for each term in the form fn+c, where d is the common difference. Once you’ve determined the common difference, you can calculate the value ofcby substituting 1fornand the first term in the series fora1 into the equation. Example 1: The arithmetic sequence 1,5,9,13,17,21,25 is an arithmetic series with a common difference of four.

  • For the thenthterm, we substituten=1,a1=1andd=4inan=dn+cto findc, which is the formula for thenthterm.
  • As an example, the arithmetic sequence 12-9-6-3-0-3-6-0 is an arithmetic series with a common difference of three.
  • It is important to note that, because the series is decreasing, the common difference is a negative number.) To determine the next3 terms, we just keep subtracting3: 6 3=9 9 3=12 12 3=15 6 3=9 9 3=12 12 3=15 As a result, the next three terms are 9, 12, and 15.
  • As a result, the formula for the fifteenth term in this series isan=3n+15.

Exemple No. 3: The number series 2,3,5,8,12,17,23,. is not an arithmetic sequence. Differencea2 is 1, but the following differencea3 is 2, and the differencea4 is 3. There is no way to write a formula in the form of forman=dn+c for this sequence. Geometric sequences are another type of sequence.

Arithmetic Sequence: Formula & Definition – Video & Lesson Transcript

Afterwards, the th term in a series will be denoted by the symbol (n). The first term of a series is a (1), and the 23rd term of a sequence is the letter a (1). (23). Parentheses will be used at several points in this course to indicate that the numbers next to thea are generally written as subscripts.

Finding the Terms

This will be represented by the letter A for the th term in the series (n). The first term of a series is a (1), and the 23rd term of a sequence is the letter a (3). (23). Parentheses will be used at several points in this course to indicate that the numbers adjacent to theaare normally written as subscripts.

Finding then th Term

Consider the identical sequence as in the preceding example, with the exception that we must now discover the 33rd word oracle (33). We may utilize the same strategy as previously, but it would take a long time to complete the project. We need to come up with a way that is both faster and more efficient. We are aware that we are starting with ata (1), which is a negative number. We multiply each phrase by 5 to get the next term. To go from a (1) to a (33), we’d have to add 32 consecutive terms (33 – 1 = 32) to the beginning of the sequence.

To put it another way, a (33) = -3 + (33 – 1)5.

a (33) = -3 + (33 – 1)5 = -3 + 160 = 157.

Then the relationship between the th term and the initial terma (1) and the common differencedis provided by:

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