We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is **a linear function** because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function.

Contents

- 1 What type of equation is an arithmetic sequence?
- 2 Are linear functions arithmetic sequences?
- 3 Is every arithmetic sequence a function?
- 4 What is an in arithmetic sequence?
- 5 What type of function is a geometric sequence?
- 6 How can you say that a sequence is an arithmetic sequence?
- 7 What is the difference between linear and arithmetic sequence?
- 8 Are all linear functions arithmetic functions?
- 9 How do you know if a sequence is a function?
- 10 Is an arithmetic sequence discrete or continuous?
- 11 What is an explicit function?
- 12 7.2 – Arithmetic Sequences
- 13 Common Difference
- 14 General Term
- 15 Partial Sum of an Arithmetic Sequence
- 16 Arithmetic Sequences – Precalculus
- 17 Introduction to Arithmetic Progressions
- 18 Formulas for Arithmetic Sequences
- 19 Using Explicit Formulas for Arithmetic Sequences
- 19.1 sequences as linear functions
- 19.2 A General Note: Explicit Formula for an Arithmetic Sequence
- 19.3 How To: Given the first several terms for an arithmetic sequence, write an explicit formula.
- 19.4 Example: Writing then th Term Explicit Formula for an Arithmetic Sequence
- 19.5 Try It
- 19.6 A General Note: Recursive Formula for an Arithmetic Sequence
- 19.7 How To: Given an arithmetic sequence, write its recursive formula.
- 19.8 tip for success
- 19.9 Example: Writing a Recursive Formula for an Arithmetic Sequence
- 19.10 How To: Do we have to subtract the first term from the second term to find the common difference?
- 19.11 Try It

- 20 Find the Number of Terms in an Arithmetic Sequence
- 21 Solving Application Problems with Arithmetic Sequences
- 22 Sequences as Functions – Explicit Form- MathBitsNotebook(A1
- 23 Sequences Basic Information – MathBitsNotebook(A1
- 24 Arithmetic Sequences
- 25 Arithmetic and Geometric Sequences

## What type of equation is an arithmetic sequence?

An arithmetic sequence is a sequence in which the difference between each consecutive term is constant. An arithmetic sequence can be defined by an explicit formula in which a_{n} = d (n – 1) + c, where d is the common difference between consecutive terms, and c = a_{1}.

## Are linear functions arithmetic sequences?

Arithmetic Sequence vs Linear Function The difference between arithmetic sequence and linear function is that an arithmetic sequence is a sequence of numbers increasing or decreasing with a constant difference whereas a linear function is a polynomial function.

## Is every arithmetic sequence a function?

Arithmetic sequences are linear functions. While the n-value increases by a constant value of one, the f (n) value increases by a constant value of d, the common difference. The rate of change is a constant “d over 1”, or just d. Geometric sequences are exponential functions.

## What is an in arithmetic sequence?

Sequences with such patterns are called arithmetic sequences. In an arithmetic sequence, the difference between consecutive terms is always the same. For example, the sequence 3, 5, 7, 9 is arithmetic because the difference between consecutive terms is always two.

## What type of function is a geometric sequence?

Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.

## How can you say that a sequence is an arithmetic sequence?

An arithmetic sequence is a list of numbers with a definite pattern. If you take any number in the sequence then subtract it by the previous one, and the result is always the same or constant then it is an arithmetic sequence.

## What is the difference between linear and arithmetic sequence?

Number sequences are sets of numbers that follow a pattern or a rule. If the rule is to add or subtract a number each time, it is called an arithmetic sequence. A sequence which increases or decreases by the same amount each time is called a linear sequence.

## Are all linear functions arithmetic functions?

Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.

## How do you know if a sequence is a function?

Definition: A sequence is a function whose domain is the set of natural numbers or a subset of the natural numbers. We usually use the symbol an to represent a sequence, where n is a natural number and an is the value of the function on n.

## Is an arithmetic sequence discrete or continuous?

Are arithmetic sequences and geometric series continuous or discrete? Discrete, values between the terms if the sequence are not included in the sequence.

## What is an explicit function?

: a mathematical function containing only the independent variable or variables —opposed to implicit function.

## 7.2 – Arithmetic Sequences

An arithmetic sequence is a succession of terms in which the difference between consecutive terms is a constant number of terms.

## Common Difference

The common difference is named as such since it is shared by all subsequent pairs of words and is thus referred to as such. It is indicated by the letter d. If the difference between consecutive words does not remain constant throughout time, the sequence is not mathematical in nature. The common difference can be discovered by removing the terms from the sequence that are immediately preceding them. The following is the formula for the common difference of an arithmetic sequence: d = an n+1- a n

## General Term

A linear function is represented as an arithmetic sequence. As an alternative to the equation y=mx+b, we may write a =dn+c, where d is the common difference and c is a constant (not the first term of the sequence, however). Given that each phrase is discovered by adding the common difference to the preceding term, this definition is a k+1 = anagrammatical definition of the term “a k +d.” For each phrase in the series, we’ve multiplied the difference by one less than the number of times the term appears in the sequence.

For the second term, we’ve just included the difference once in the calculation.

When considering the general term of an arithmetic series, we may use the following formula: 1+ (n-1) d

## Partial Sum of an Arithmetic Sequence

A series is made up of a collection of sequences. We’re looking for the n th partial sum, which is the sum of the first n terms in the series, in this case. The n thpartial sum shall be denoted by the letter S n. Take, for example, the arithmetic series. S 5 = 2 + 5 + 8 + 11 + 14 = S 5 = 2 + 5 + 8 + 11 + 14 = S 5 The sum of an arithmetic series may be calculated in a straightforward manner. S 5 is equal to 2 + 5 + 8 + 11 + 14 The secret is to arrange the words in a different sequence. Because addition is commutative, altering the order of the elements has no effect on the sum.

- 2*S 5= (2+14) + (5+11) + (8+8) + (11+5) + (14+2) = (2+14) + (5+11) + (8+8) + (11+5) + (14+2) Take note that each of the amounts on the right-hand side is a multiple of 16.
- 2*S 5 = 5*(2 + 14) = 2*S 5 Finally, divide the total item by two to obtain the amount, not double the sum as previously stated.
- This would be 5/2 * (16) = 5(8) = 40 as a total.
- The number 5 refers to the fact that there were five terms, n.
- In this case, we added the total twice and it will always be a 2.
- Another formula for the n th partial sum of an arithmetic series is occasionally used in conjunction with the previous one.

It is produced by putting the generic term formula into the previous formula and simplifying the result. Instead of trying to figure out the n thterm, it is preferable to find out what it is and then enter that number into the formula. In this case, S = n/2 * (2a 1+ (n-1) d).

### Example

When a sequence is added up, you get a series of numbers. Finding the n th partial sum, or the total of the first n terms of the series, is what we’re looking for. The n thparticipant sum will be denoted by the letter S. Take, for example, the arithmetic series: In S 5, the sum of the first five numbers is two plus five plus eight plus eleven plus fourteen. An arithmetic series may be added together in a straightforward manner. The sum of two and five is eight and eleven and fourteen. Change the order of the phrases in the sentence is the key to solving the problem.

- 2+2=14+11+8+5+2=5+2=14+11+8+5+2=5 Now, combine the results of the two equations.
- Instead of writing 16 (the sum of the first and final terms) five times, we may express it as 5 * 16 or 5 * (2 + 14), which is the same as 5 * (first and last terms) plus 5.
- Once you’ve finished, divide the total by two to obtain the amount, not twice the sum.
- To calculate this, multiply 5/2 by (16), which is 5(8).
- Five words were used in all, therefore the number 5.
- In this case, we added the total twice and it will always equal a 2.
- Another formula for the n th partial sum of an arithmetic series is sometimes employed, and it is denoted by the symbol To produce it, substitute the formula for the general term into the previous formula, which has been simplified.
- (2) 2a 1+ (d-1) d = S n = 2a 2 * (2a 1+ (d-1) d

## Arithmetic Sequences – Precalculus

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

### 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.

Specific types of sequences that will allow us to compute depreciation, such as the value of a truck’s worth, will be discussed in detail in this section.

### 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?
- It is referred to be anarithmetic sequence in the context of the vehicle in the example since the values change by a fixed amount each year.
- –3,400 is the common difference between the two sequences in this case study.
- The constant difference in this situation is three.
- Number Sequences in Arithmetic An anarithmetic sequence is a series that has the characteristic that the difference between any two consecutive words is always the same.
- ” Given that and is the initial term of an arithmetic series, and is the common difference between them, the sequence will be as follows: Differentiating between the two types of people Is each of these sequences mathematical in nature?

- The series is not arithmetic because there is no common difference between the elements
- 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.

- To determine the second term, add the common difference to the first term
- And so on. To determine the third term, add the common difference to the second term
- This will give you the third term. Make sure to keep going until you’ve found all of the needed keywords
- 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

Arithmetic Sequences in the Form of Terms Make a list of the first five terms of the arithmetic sequence with and without Subtracting 3 from any number is equivalent to adding 3. Starting with the first phrase, deduct 3 from each word to arrive at the next term in the progression. First and foremost, there are five terms: Analysis To be predicted, the sequence’s diagram is made up of points on a line, as seen in Figure 1. (Figure). Write down the first five terms of the arithmetic series that begins with and ends withand Arithmetic Sequences in the Form of Terms Find out what you’ve got.

In this case, we know that the fourth term equals 14 and that it has the form.

By adding the common difference to the fourth term, we may find the fifth term.

The tenth term might be obtained by multiplying the common difference by the first term nine times, or by using the following equation:

- 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

Making a Recursive Formula for an Arithmetic Sequence is a difficult task. Write a recursive formula for the arithmetic series in the following format: The first term is defined as follows. It is possible to calculate the common difference by subtracting the first term from the second term. In the recursive formula for arithmetic sequences, substitute the initial term and the common difference in place of the first term. Analysis We can observe that the common difference is the slope of the line generated when the terms in the sequence are graphed, as illustrated in the figure below (Figure).

Is it necessary to deduct the first term from the second term in order to obtain the common difference between the two?

We can take any phrase in the sequence and remove it from the term after it.

Create a recursive formula for the arithmetic sequence using the information provided.

### Using Explicit Formulas for Arithmetic Sequences

Formalizing an Arithmetic Sequence as a Recursive Formula Write a recursive formula for the arithmetic series in a separate document. The first phrase is denoted by the letters By subtracting the first term from the second term, we may get the common difference between them. Make an addition to the recursive formula for arithmetic sequences and substitute the beginning term and the common difference in place of them. Analysis We can observe that the common difference is the slope of the line generated when the terms in the sequence are graphed, as shown in the illustration (Figure).

When calculating the common difference, do we need to deduct the first from the second term?

The next phrase can be subtracted from any term in the series.

For the arithmetic series, write out the recursive formula for it.

- Identify the commonalities and differences
- 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 frequently 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 seen in (Figure) that the slope of this sequence is 10 and that the vertical intercept is 10.

#### 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. The number of terms in the finite arithmetic sequence has to be determined. 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.

- Create a method for calculating the child’s weekly stipend over the course of a year
- What will be the child’s allowance when he reaches the age of sixteen

- 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

A lady chooses to go for a 10-minute run every day this week, with the goal of increasing the length of her daily exercise by 4 minutes each week after that. Create a formula to predict the timing of her run after n weeks has passed. In eight weeks, how long will her daily run last on average? The formula is, and it will take her 42 minutes to complete the task.

### 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.

- The first term is 3, the most common difference is 4, and the fifth term is 3.
- 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.
- To complete the following problems, you must discover the first term from an arithmetic series given two terms.
- Make use of the recursive formula in order to write the first five terms of the arithmetic sequence in each of the following tasks.
- 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.
- Find the fourteenth term.
- For the following problems, write the first five terms of the arithmetic sequence using the explicit formula that was provided.

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. Six words are contained within the sequence of letters.

#### 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

In order to complete the following activities with an arithmetic sequence and a graphing calculator, follow the methods listed below:

- 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?

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

## 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:

- 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

Allowing ‘l’ to signify the n thterm of the series and S n to represent the sun of the first n terms of the series a, (a+d), (a+2d),., a+(n-1)d S n = a 1 plus a 2 plus a 3 plus .a n-1 plus a n S n= a + (a + d) + (a + 2d) +. + (l – 2d) + (l – d) + l. + (l – 2d) + (l – d) + l. + (l – 2d) + (l – d) + l. (1) When we write the series in reverse order, we obtain S n= l + (l – d) + (l – 2d) +. + (a + 2d) + (a + d) + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a … (2) Adding equations (1) and (2) results in equation (2).

+ (a + l) + (a + l) + (a + l) +.

(3) As a result, the formula for calculating the sum of a series is S n= (n/2)(a + l), where an is the first term of the series, l is the last term of the series, and n is the number of terms in the series.

d S n= (n/2)(a + a + (n – 1)d)(a + a + (n – 1)d) S n= (n/2)(2a + (n – 1) x d)(n/2)(2a + (n – 1) x d) Observation: The successive terms in an Arithmetic Progression can alternatively be written as a-3d, a-2d, a-d, a, a+d, a+2d, a+3d, and so on.

### Sample Problems on Arithmetic Progressions

Allowing ‘l’ to signify the n thterm of the series and S n to represent the sun of the first n terms of the series a, (a+d), (a+2d),., a+(n-1)d S n= a 1 plus a 2 plus a 3 plus .a n-1 plus a n + (l – 2d)+ (l – d)+ l.+ (l – 2d) + (l – 2d) + l. + (l – 2d) + (l – 2d) + l. + (l – 2d) + (l – 2d) + l. + (l – 2d) + (l – 2d) + l. (1) Sn = l + (l – 2d) +. + (a + 2d) + a = a + a + a + a + a + a = l+ (l – 2d) + a = a + a + a = a + a + a = a + a + a = a + a = a + a = a + a = a + a = a + … (2) Equations (1) and (2) are combined to form the second equation.

plus (a+ l) plus((a+ l) plus (a+ l) plus (a + l) + (a + l) +.

2 As an example, consider the following equation: Sn=a+ln(a).

(3) In this case, the formula to get the sum of a series is S n= (n/2)(a + l), where an is the first term in the series, l is the last term in the series, and n is the number of terms in the series.

d In the case of S n, (n/2)(a + a + (n – 1)d) is the value of S n. In the case of S n= (n/2)(2a + (n – 1) x d), the formula is Observation: The successive terms in an Arithmetic Progression can alternatively be written as a-3d, a-2d, a-d, a, a+d, a+2d, a+3d, and so on.

## Formulas for Arithmetic Sequences

- Create a formal formula for an arithmetic series using explicit notation
- Create a recursive formula for the arithmetic series using the following steps:

## 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. = +dleft = +dright 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.

- Considering that the average difference is 50, the series represents a linear function with an associated slope of 50.
- You may also get the they-intercept by graphing the function and calculating the point at which a line connecting the points would intersect the vertical axis, as shown in the example.
- When working with sequences, we substitute _instead of y and ninstead of n.
- Using 50 as the slope and 250 as the vertical intercept, we arrive at this equation: = -50n plus 250

### sequences as linear functions

So far in this module, we’ve examined a number of graphs containing sequence terms. It may be more helpful to think of an arithmetic sequence as a linear function of the formmy=mx+b or, in sequence notation,a n=dn+a 0where each point on the graph is of the formleft(n, a nright)and the common difference gives us the slope of the line if you think of it as a linear function of the formmy=mx+b. Remember that if we put the value 0 in the explicit forma n=a 1+d(n-1), we get the value a 0=a 1-d in the resulting statement.

To create an explicit formula for an arithmetic series, we do not need to identify the vertical intercept of the sequence.

### A General Note: Explicit Formula for an Arithmetic Sequence

For the textterm of an arithmetic sequence, the formula = +dleft can be used to express it explicitly.

### How To: Given the first several terms for an arithmetic sequence, write an explicit formula.

- Find the common difference between the two sentences, – To solve for = +dleft(n – 1right), substitute the common difference and the first term into the equation

### Example: Writing then th Term Explicit Formula for an Arithmetic Sequence

Create an explicit formula for the arithmetic sequence.left 12text 22text 32text 42text ldots right 12text 22text 32text 42text ldots

### Try It

For the arithmetic series that follows, provide an explicit formula for it. left 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. The initial term in every recursive formula must be specified, just as it is with any other formula. the beginning of the sentence = +dnge 2 the finish of the sentence

### A General Note: Recursive Formula for an Arithmetic Sequence

For the following arithmetic sequence, provide an explicit formula for it. left An arecursive formula is used to define some arithmetic sequences in terms of the previous phrase. When finding the terms of the sequence, the formula gives an algebraic procedure that may be applied. A recursive formula allows us to locate any term in an arithmetic series by utilizing a function of the term that came before it in the sequence. It is calculated by adding up each term’s previous term and the common difference between them.

Whenever a recursive formula is used, it is necessary to provide the initial term.

### How To: Given an arithmetic sequence, write its recursive formula.

- 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

### tip for success

When describing an arithmetic series in which the initial term is known, it is possible to use either the explicit or the recursive form of the expression. Use the examples in this section to get some practice with both of them.

### Example: Writing a Recursive Formula for an Arithmetic Sequence

Write a recursive formula for the arithmetic series in the following format: left

### How To: Do we have to subtract the first term from the second term to find the common difference?

No. We can take any phrase in the sequence and remove it from the term after it. Generally speaking, though, it is more customary to subtract the first from the second term since it is frequently the quickest and most straightforward technique of determining the common difference.

### Try It

Create a recursive formula for the arithmetic sequence using the information provided. left

## Find the Number of Terms in an 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.

### How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

- Find the common differences between the two
- To solve for = +dleft(n – 1right), substitute the common difference and the first term into the equation Fill in the blanks with the final word from and solve forn

### tip for success

Remember that we only only one equation in one unknown in order to find the answer. Using the explicit form of an arithmetic sequence,a n=a 1+d(n-1), we may solve for the missing component by substituting known values for all but one of the components. In this scenario, we are told that a certain finite series is arithmetic, and we know the first terma 1 and the last terma n of the sequence in question. Using any two consecutive phrases that we are provided, we are able to determine the common difference between the terms.

### Example: Finding the Number of Terms in a Finite Arithmetic Sequence

The number of terms in the infinite arithmetic sequence is to be determined. left

### Try It

The number of terms in the finite arithmetic sequence has to be determined. 11 text 16 text. text 56 right 11 text 16 text 16 text 56 text 56 text 56 Following that, we’ll go over some of the concepts that have been introduced so far concerning arithmetic sequences in the video lesson that comes after that.

## Solving Application Problems with Arithmetic Sequences

In many application difficulties, it is frequently preferable to begin with the term instead of_ as an introductory phrase.

When solving these problems, we make a little modification to the explicit formula to account for the change in beginning terms. The following is the formula that we use: = +dn = = +dn

### Tip for success

Keep in mind as you work through the examples and practice problems that an arithmetic sequence can be represented as a linear function with inputn, outputa n, and a common difference (or slope) ofd. See the red box SEQUENCES AS LINEAR FUNCTIONS at the beginning of this section for a derivation and explanation of this formula.

### Example: Solving Application Problems with Arithmetic Sequences

Every week, a kid under the age of five receives a $1 stipend from his or her parents. His parents had promised him a $2 per week rise on a yearly basis.

- Create a method for calculating the child’s weekly stipend over the course of a year
- What will be the child’s allowance when he reaches the age of sixteen

### Try It

Calculate the child’s weekly allowance using a formula for a certain year. Which amount will be the child’s allowance when he reaches the age of sixteen?

## Sequences as Functions – Explicit Form- MathBitsNotebook(A1

We saw inSequences – Basic Information, that sequences can be expressed in various forms.This page will look at one of those forms, the explicit form.Certain sequences (not all) can be defined (expressed) as an “explicit” formula.An explicit formulawill create a sequence usingn, the numberlocation of each term. If you can find an explicit formula for a sequence, you will be able to quickly and easily find any term in the sequence simply by replacingnwith the number of the term you seek.Anexplicit formuladesignates thenthterm of the sequence,as an expression ofn(wheren= the term’s location). It defines the sequence as a formula in terms ofn. It may be written in either subscript notationan, or in functional notation,f(n).Sequence: . Find an explicit formula. This example is anarithmetic sequence(the same number, 5, is added to each term to get tothe next term). |

Term Number | Term | Subscript Notation | Function Notation |

1 | 10 | a1 | f(1) |

2 | 15 | a2 | f(2) |

3 | 20 | a3 | f(3) |

4 | 25 | a4 | f(4) |

5 | 30 | a5 | f(5) |

6 | 35 | a6 | f(6) |

n | a n | f(n) |

This sequence is graphed in the first quadrant. Remember that the domain consists ofthe natural numbers, and the range consists of the terms of the sequence.Notice that this sequence has alinear appearance.The rate of change between each of the points is “5over 1”.While thenvalue increases by a constant value of one, thef(n) value increases by a constant value of 5(for this graph).Willarithmetic sequences be linear functions? |

It is easy to see that the explicit formula works once you are giventhe formula. Unfortunately, it is not always easy to come up withexplicitformulas,when all you have is a list of the terms.If your sequence is arithmetic, it will help if you look at the pattern of what is happening in the sequence. | |

Explicit formula:f(n) = 10 + 5(n- 1) | If you compare the term number with how many times the common difference, 5, is added, you will see a pattern for an explicit formula: |

For the purpose of summarizing the process of developing an explicit formula for an arithmetic series, consider the following: 1st, determine whether the sequence is arithmetic (does each phrase add or remove the same amount from the previous term) 2.Identify the common point of disagreement. (It is the number you add or subtract from the total.) 3.Create an explicit formula using the pattern of the first term added to the product of the common difference and one less than the term number and one more than the term number.

a n=a1+d(n- 1) | a n= then thterm in the sequencea1= the first term in the sequencen= the term numberd= the common difference. | |

first term = 10, common difference = 5explicit formula:a n= 10 + 5(n- 1)= 10 + 5 n- 5 = 5 + 5 nor 5 n +5 |

Using the explicit formula, identify the 100 th term in this series of numbers. In the explicit formula, 100 has been substituted for: f(n) = 10 + 5 = f(n) (n- 1) 100 = 10 + 5(100 – 1) = 10 + 5(99) = 10 + 495 =505 100 + 5(99) = 10 + 495 =505 THE ANSWERSequence is:. Look for a formula that is explicit. The following is an example of an ageometric sequence (the same number, 2, is multiplied times each term to get tothe next term).

Term Number | Term | Subscript Notation | Function Notation |

1 | 3 | a1 | f(1) |

2 | 6 | a2 | f(2) |

3 | 12 | a3 | f(3) |

4 | 24 | a4 | f(4) |

5 | 48 | a5 | f(5) |

6 | 96 | a6 | f(6) |

n | a n | f(n) |

Notice that this sequence has anexponential appearance.It may be the case with geometric sequences that the graph will increase (or decrease). The rate of change will increase (or decrease) as the value ofnincreases (it is not constant). Will such geometric sequences be exponential functions? |

Again, it is easy to see that a given explicit formula works. The problem is coming up with a formula when all you are given is a list of the terms.If your sequence is geometric, it will help if you look at the pattern of what is happening in the sequence, in a manner similar to what we examined in the arithmetic sequence. | |

Explicit formula:f(n) = 32n -1 | If you compare the term number with the powers ofthe common difference, 2,you will see a pattern for an explicit formula: |

This is a brief summary of the steps involved in constructing an explicit formula for a geometric sequence: 1.Check to see if the sequence is geometric (i.e., do you multiply or divide by the same number from one term to the next?) 2. Calculate the common ratio. (The result of multiplying or dividing a number.) Develop an explicit formula utilizing the pattern of the first term multiplied by the common ratio raised to a power of one less than the term number, and then evaluate the expression.

a n= then thterm in the sequencea1= the first term in the sequencen= the term numberr= the common ratio | ||

first term = 3, common ratio = 2explicit formula:a n= 32n -1 |

Now that you have the exact formula, you must determine the 9 thterm in this series of numbers. In the explicit formula, the number 9 has been substituted: f(n) = 32n-1 f(9) = 32 9-1= 32 8= 3256 = 768 f(n) = 32n-1 f(9) = 32 9-1= 32 8= 3256 = 768 f(n) = 32n-1 f(9) = 32 9-1= 32 8= 3256 = 768 Sequence: This example is neither an arithmetic nor a geometric sequence in the traditional sense.

Seeing the pattern for an explicit formula foran arithmetic sequence or a geometric sequence will be easy ascompared to findingexplicit formulas for sequences that do not fall into these categories.The sequence shown in this example isa famous sequence called theFibonacci sequence. | |

The Fibonacci sequence is famous as being seen in nature (leaf arrangements, bracts of pine cones, scales of pine cones, sunflowers, flower petals, Nautilus shells, grains of wheat, coniferous trees, bee hives, and even single cells). It isreferred to as Nature’s numbering system. The Fibonacci sequence is alsoassociated with the golden ratio (1.61803), which can be seen in the ratio of two successive terms of the Fibonacci sequence (as the Fibonacci numbers grow). | |

Is there apattern in the Fibonacci sequence?Yes. After the first two terms, each term is thesum of the previous two terms.Is there an explicit formula forthe Fibonacci sequence?Yes. |

Explicit Formula: | (Don’t panic!You will not be asked to find explicit formulas of this difficulty level.But you may be asked to “use” a more difficult given formula.) |

What about the graph of the Fibonacci sequence?As seen at the right, when graphed, the Fibonacci sequence takes on the appearance of an exponential graph. While it is not truly exponential, the Fibonacci sequence can be “modeled” with an exponential function. With the sequence’s connection to the golden ratio, it canbe “modeled” by an exponential function with 1.6 as the base,f(x) = 1.6x. (This is a “model”, not an exact formula match.) |

## Sequences Basic Information – MathBitsNotebook(A1

Asequenceis an ordered list.It is a function whose domain is the natural numbers. |

Sequence: | 1, | 5, | 9, | 13, | 17, | 21, | . |

Notation for termsof the sequence: | a1 | a2 | a3 | a4 | a5 | a6 |

Information about sequences is as follows:

Each number in a sequence is called aterm,anelementor amember. | |

Terms are referenced in a subscripted form (indexed), where thenatural number subscripts, refer to thelocation (position) of the term in the sequence. The first termis denoteda1, the second terma2, and so on.Thenthterm isan. | |

The terms in a sequencemay, or may not, have a patternor related formula.Example:the digits ofπform a sequence, but do not have a pattern. | |

Asubscripted form of a sequenceis represented bya1,a2,a3,.an,. | |

Afunctionalform of a sequenceis represented byf (1), f (2), f (3),., f (n),. | |

Sequences arefunctions. | |

Thedomainof a sequence consists of the natural (counting) numbers 1, 2, 3, 4,. | |

Therangeof a sequence consists of the terms of the sequence. | |

Whengraphed, a sequence is a scatter plot, a series of dots.(Do not connect the dots). | |

Thesumof the terms of a sequence is called aseries. |

Sequences can be represented in a variety of ways, including:

Term Number | Term | Subscript Notation | Function Notation |

1 | 1 | a1 | f(1) |

2 | 5 | a2 | f(2) |

3 | 9 | a3 | f(3) |

4 | 13 | a4 | f(4) |

5 | 17 | a5 | f(5) |

6 | 21 | a6 | f(6) |

n | a n | f(n) |

(list)Subscripted notation: an= 4 n- 3(explicit form)a1= 1; an=an-1+ 4(recursive form)Functional notation: f(n) = 4 n- 3(explicit form)f(1) = 1;f(n) =f(n- 1) + 4 (recursive form)Note:Not all functions can be defined by an explicit and/or recursive formula. |

Sequences can take the following forms:

Afinite sequencecontains a finite number of terms (a limited number of terms) which can be counted. Example:(it starts and it stops) | |

Aninfinite sequencecontains an infinite number of terms(terms continue without end) which cannot be counted. Example:(it starts but it does not stop, as indicated by the ellipsis.) |

Sequences can be expressed (defined) in a variety of ways.

A sequence may appear as alist(finite or infinite): Examples:and Listing makes it easy to see any pattern in the sequence.It will be the only option should the sequence have no pattern. | |

A sequence may appear as anexplicit formula. Anexplicit formuladesignates thenthterm of the sequence,an, as an expression ofn(wheren= the term’s location).Example:can be writtenan= 4 n- 3. (a formula in terms of n)Read more atSequences as Functions – Explicit | |

A sequence may appear as arecursive formula. A recursive formula designates the starting term,a1, and thenthterm of the sequence,an, as an expression containing the previous term (the term before it),an-1.Example:can be writtena1= 1; an=an-1+ 4.(two-part formula in terms of the preceding term)Read more atSequences as Functions – Recursive. |

Graphing Sequences (Graphing Sequences):

Sequence:Sequences arefunctions. They pass the vertical line test for functions.Thedomainconsists of the natural numbers, and therangeconsists of the terms of the sequence.The graph will be in thefirst quadrantand/or thefourth quadrant(if sequence terms are negative). | |

Arithmetic sequencesarelinear functions.While then -value increases by a constant value of one, thef(n) value increases by a constant value ofd, the common difference.The rate of change is a constant” dover 1″, or justd.Geometric sequencesareexponential functions.While then -value increases by a constant value of one, thef(n) value increases bymultiples ofr, the common ratio.The rate of change is not constant, but increases or decreases over the domain. |

When dealing with sequences, you should constantly be on the lookout for popular sequence patterns, such as those indicated in the table below. Keep in mind, however, that while all sequences have an order, they may or may not have a pattern to them.

Arithmetic Sequence:(where youadd(orsubtract) thesame valueto get from one term to the next.)If a sequence adds a fixed amount from one term to the next, it is referred to as an arithmetic sequence. The number added to each term is constant (always the same) and is called thecommon difference,d. The scatter plot of this sequence will be alinear function. | |

Geometric Sequence:(where youmultiply(ordivide) thesame valueto get from one term to the next.) If a sequence multiplies a fixed amount from one term to the next, it is referred to as a geometric sequence. The number multiplied is constant (always the same) and is called thecommon ratio,r. The scatter plot of this sequence will be anexponential function. | |

Doubting Thomas wonders how we can know, for sure, that a sequence such as2, 4, 6, 8,. is an arithmetic sequence. His theory is that there could be many other possible patterns, such as:2, 4, 6,8,2, 4, 6,8,.(repeating 4 terms is his pattern). Yes, Thomas is correct.Without a specification in the problem, there is the possibility of more than one pattern in most sequences.The person creating the sequence may have been thinking of a different pattern than what you see when you look at the sequence. InAlgebra 1, if in doubt, first look for arithmetic or geometric possibilities. |

Note:Theindexing(subscripts) used for sequences can begin with 0 or any positive integer. The most popular indexing, however,begins with 1 so the index can also represent the position of the termin the sequence. Unless otherwise stated, this site willstart indexes at 1. | Note:Computer programming languagessuch as C, C++ and Java, refer tothe startingposition in an array with a subscript of zero. Programmers must remember that a subscript of 3 refers to the 4 thelement, not the 3 rdelement, in the array. |

## 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 and Geometric Sequences

The wordarithmetic (in this context) is pronounced air-ith-ME-tic, which means that the accent is on the third syllable when you are talking about anarithmetic sequence. Each term in an arithmetic series is equal to the preceding term plus (or minus) a constant, with the exception of the first term. It is an arithmetic sequence to have the numbers$,4$, $,7$, $,10$, $,13$, and $ldots$ as the first four digits. The most often seen difference is $,3$. To proceed from one phrase to the next, you must keep adding $,3$.

- Even arithmetic sequences have this unique property: equal changes in the input (for example, going from term to term) result in equal changes in the output (as in the case of arithmetic sequences) (determined by the common difference).
- The collection of points (‘dots’) displayed below represents the graph of the sequence$,4$, $,7$, $,10$, $,13$, $,ldots$, $,ldots$, $,ldots$, $,ldots$, $,ldots$, $,ldots$, $,ldots$, $,ldots$, $ It is possible to get an output of $,4$ by using $,1$ as an input (for the first term in a series).
- When the input is $,3$ (for the third term in the sequence), the output is $,10$, and so on until the sequence is completed.
- The most often encountered difference is $,-2,$.

The graph of the sequence $,10,$, $,8,$, $,6,$, $,4,$, $,ldots,$ is given below: $,10,$, $,8,$, $,6,$, $,4,$, $,ldots,$, $,ldots,$, $,ldots,$, $,ldots,$, $,ldots DEFINITION geometric sequence, common ratio, and so forth When the sequence is of the form$,u n = rcdotu_ n,$, it is called an ageometric sequence.

If you look at it like this, each term in a geometric series is equal to the preceding term multiplied (or divided) by a constant.

You must keep multiplying by $,2$ in order to proceed from term to term.

In addition to this specific trait, geometric sequences have another: equivalent changes in the input (for example, shifting from term to term) cause the output to be repeatedly multiplied by a constant (determined by the common ratio).

The following graph depicts the sequence$,3$, $,6$, $,12$, $,24$, $,ldots$, $,3$, $,6$, $,3$, $,3$, $,3$, $,3$, $,3$, $,3$, $,3$, $,3$, $,3$, $,3$, $,3$, $,3$, $,3$, $, THE FOLLOWING IS AN EXAMPLE (geometric sequence): The geometric sequence$,100,$, $,50,$, $,25,$, $,12.5,$, $,ldots,$ is composed of the numbers$,100,$, $,50,$, $,25,$, $,12.5,$, $,ldots,$.

It is necessary to keep multiplying by $,frac,$ (i.e.

The following is the graph of the sequence$,100,$, $,50,$, $,25,$, $,12.5,$, $,ldots,$: $,100,$, $,50,$, $,25,$, $,12.5,$, $,ldots,$ By completing the activity at the bottom of this page, you will be able to master the concepts presented in this part.

After you’ve finished practicing, continue on to the following topics: Loans and Investments.