Answer: In the arithmetic sequence a_{14}= -75 and a_{26} = -123, the recursive formula which defines the sequence is **-19 – 4n**.

Contents

- 1 What is a recursive formula for the arithmetic sequence?
- 2 Which formula defines the sequence?
- 3 What is a recursive formula example?
- 4 What is a recursive definition in math?
- 5 Is arithmetic recursive?
- 6 Introduction to Arithmetic Progressions
- 7 Formulas for Arithmetic Sequences
- 8 Using Explicit Formulas for Arithmetic Sequences
- 8.1 A General Note: Explicit Formula for an Arithmetic Sequence
- 8.2 How To: Given the first several terms for an arithmetic sequence, write an explicit formula.
- 8.3 Example: Writing then th Term Explicit Formula for an Arithmetic Sequence
- 8.4 Try It
- 8.5 A General Note: Recursive Formula for an Arithmetic Sequence
- 8.6 How To: Given an arithmetic sequence, write its recursive formula.
- 8.7 Example: Writing a Recursive Formula for an Arithmetic Sequence
- 8.8 How To: Do we have to subtract the first term from the second term to find the common difference?
- 8.9 Try It

- 9 Find the Number of Terms in an Arithmetic Sequence
- 10 Solving Application Problems with Arithmetic Sequences
- 11 Contribute!
- 12 Writing the Terms of a Sequence Defined by a Recursive Formula
- 12.1 A General Note: Recursive Formula
- 12.2 QA
- 12.3 How To: Given a recursive formula with only the first term provided, write the firstnterms of a sequence.
- 12.4 Example 5: Writing the Terms of a Sequence Defined by a Recursive Formula
- 12.5 Solution
- 12.6 Try It 7
- 12.7 How To: Given a recursive formula with two initial terms, write the firstnterms of a sequence.
- 12.8 Example 6: Writing the Terms of a Sequence Defined by a Recursive Formula
- 12.9 Solution
- 12.10 Try It 8

- 13 Using Factorial Notation
- 14 Sequences as Functions – Recursive Form- MathBitsNotebook(A1
- 15 Recursive Formula (Explained w/ 25 Step-by-Step Examples!)
- 16 What Is A Sequence
- 17 Recursive Formula Definition
- 18 Example
- 19 Recursive Formulas For Sequences
- 20 Example
- 21 Example
- 22 Example
- 23 Summary
- 24 Video Tutorial w/ Full LessonDetailed Examples
- 25 Arithmetic Sequence: Formula & Definition – Video & Lesson Transcript
- 26 Finding the Terms
- 27 Finding then th Term

## What is a recursive formula for the arithmetic sequence?

The arithmetic sequence recursive formula is: an=an−1+d. where, an = n^{th} term of the arithmetic sequence.

## Which formula defines the sequence?

A geometric sequence is one in which a term of a sequence is obtained by multiplying the previous term by a constant. It can be described by the formula an=r⋅an−1 a n = r ⋅ a n − 1.

## What is a recursive formula example?

an=2an−1+3 is a recursive formula because each term, an, refers back to the previous term, an−1. This equation is telling us that whatever term we want to find is equal to 2 times the previous term, plus 3. The first three terms of this sequence are: 4,11,25.

## What is a recursive definition in math?

recursive function, in logic and mathematics, a type of function or expression predicating some concept or property of one or more variables, which is specified by a procedure that yields values or instances of that function by repeatedly applying a given relation or routine operation to known values of the function.

## Is arithmetic recursive?

A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. As with any recursive formula, the first term must be given.

## 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

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.

## 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 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

What’s the most significant distinction between you and everyone else? To solve for = +dleft(n – 1right), substitute the common difference and the first term.

### 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.

The initial term in every recursive formula must be specified, just as it is with any other formula.

### 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.

In this case, if the common difference is 5, then each word is equal to the preceding term + 5. Whenever a recursive formula is used, it is necessary to provide the initial term. Beginning with the letter _, and ending with the letter _, we have the expression

### 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

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

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.

### 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

### 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

### 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

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?

## Contribute!

Do you have any suggestions about how to make this article better? We would much appreciate your feedback. Make this page more user-friendly. Read on to find out more

## Writing the Terms of a Sequence Defined by a Recursive Formula

Neptune shells, pine cones, tree branches, and a variety of other natural structures have development patterns that are composed of sequences in their formation. The sequence of leaves or branches in a plant’s layout, the number of petals on a flower, or the pattern of chambers in a nautilus shell are all examples of sequence. Their development follows the Fibonacci sequence, which is a well-known series in which each term can be determined by adding the two terms that came before it. There are a total of nine numbers in the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34.

- Each word in the Fibonacci sequence is reliant on the terms that came before it in order to function correctly.
- Instead, we express the series using an arecursive formula, which is a formula that specifies the terms of a sequence by referencing previously defined terms.
- Consider the following scenario: we know the following: Begin =3hfill =2 -1, begin =3hfill =2 -1, textnge 2hfill -1, end =3hfill -1, begin =3hfill -1, end =3hfill -1 Using the first term in the sequence, we may determine the subsequent terms in the series.
- (3right) the value of -1=5 is equal to the value of =2 the value of -1=2 is left -1=9 =2 -1=2left(9right)-1=17end -1=9 =2 -1=2left(9right)-1=17end As a result, the first four terms of the series are as follows: left 5, text 9, text 17, and right.
- Beginning with 1h, filling in 1h, filling in 1h, filling in 1h, textnge 3h, and ending with 1h, filling in 1h, textnge 3h, and ending with 1h.

For example, in order to determine the tenth term of a series, we would need to add the eighth and ninth terms together. We were previously informed that the eighth and ninth terms are 21 and 34, respectively, and that = + =34+21=55.

### A General Note: Recursive Formula

An arecursive formula is a formula that defines each phrase of a series by referencing the word that came before it (s). Recursive formulae must always include the first term (or terms) in the series in the body of the formula.

### QA

No. The Fibonacci sequence defines each term by referencing the two words that came before it, whereas many recursive formulae define each term by referencing just one prior phrase that came before it. It is only necessary to define the first word in these sequences.

### How To: Given a recursive formula with only the first term provided, write the firstnterms of a sequence.

- The first term in the formula is denoted by the letter_. Determine what this term is. This is the first of three terms. To obtain the second term,_ just insert the first term into the formula for_ and multiply by two. Solve the problem by substituting the second term into the formula to determine the third term. Solve
- Continue until you have found a solution for thentextterm.

### Example 5: Writing the Terms of a Sequence Defined by a Recursive Formula

The first five terms of the sequence formed by the recursive formula should be written down. =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9 =9

### Solution

A definition for the first term may be found in the formula. We replace the value of the preceding term with the value of the succeeding term for each subsequent phrase. begin=1 beginhfill beginhfill beginhfill begin n=2 beginhfill begin n=3 beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill begin n=3 beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginhfill beginh The first five words are as follows: left 7, text 1, text -17, text -71, and right.

Figure 6 shows a diagram of the human body.

### Try It 7

The first five terms of the sequence formed by the recursive formula should be written down. begin =2 =2 +1text nge 2end begin =2 =2 +1text nge 2end begin =2 =2 +1text nge 2end begin =2 =2 +1text nge 2end begin =2 +1text nge 2end begin

### How To: Given a recursive formula with two initial terms, write the firstnterms of a sequence.

- Locate and describe the starting term_, which is provided as part of the formula. The second term, which is presented as part of the formula, needs to be identified. In order to determine the third term, first insert the first term and then the second term into the formula for three. Evaluate
- Continue until you have analyzed the textterm
- Then stop.

### Example 6: Writing the Terms of a Sequence Defined by a Recursive Formula

The first six terms of the sequence formed by the recursive formula should be written down. starting with =1

### Solution

The first two terms have been provided. The values of the two preceding terms are used to replace and with the values of the succeeding terms. n=4hfillhfillhfillhfill =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 = These are the first six terms: text text Figure 7:

### Try It 8

You will be given the first two words. We replace and with the values of the two phrases that came before it for each subsequent term. n=4hfillhfillhfillhfillhfill =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 =3 +4 These are the first six terms:text textFigure 7:

## Using Factorial Notation

Some sequences include formulae that incorporate products of successive positive integers. These are known as positive integer products. nfactorial, abbreviated asn!, is the sum of all positive integers from one to one hundred. For example, begin4!=4cdot 3cdot 2cdot 1=24hfill 5!=5cdot 4cdot 3cdot 2cdot 1=120hfill end4!=5cdot 4cdot 3cdot 2cdot 1=120hfill end4!=5cdot 4cdot 3cdot 2cdo As an illustration, consider the formula afactorialis_ =left(n+1)right. . Using the substitution 6 forn._ =left(6+1)right, the sixth term in the sequence may be found.

+=7!=7!=7!=7!=7!=7!=7!=7!=7cdot 6cdot 5cdot 4cdot 3cdot 2cdot 1=5040 When dealing with whole numbers, remember that the factorial of every whole numbernisn(1 – 1)! As a result, we may conceive about 5! as 5cdot 4!text as well.

### A General Note: Factorial

Nfactorial is a mathematical operation that can be defined by using a recursive formula, which is shown in the following example. It is defined as follows for a positive integernas the factorial ofn, denotedn! Begin with 0!=1 and go up to 1!=1 and go left (n – 1right) ‘left’ is a slang term for “left-handed” or “left-handedness” (n – 2right) cdots to the left (2right) left(1right) text nge 2nd stanza The exceptional case0!is defined as0!=1 in the following way:

### QA

No. Factorials grow in size quite quickly—much more swiftly than even exponential functions! Whenever the output becomes too huge for your calculator to handle, it will be unable to calculate the factorial.

### Example 7: Writing the Terms of a Sequence Using Factorials

Write the first five terms of the series specified by the explicit formula_ =frac, followed by the rest of the sequence.

### Solution

n=2hfillhfillhfill_ =frac=frac=frac=frachfill n=3 hfillhfillhfill_ =frac=frac=frac=frachfill n=4hfillhfillhfill_ =frac=frac=frac=frac008hfill n=5hfillhfillhfill_ =frac=frac=frac008hfill n=6hfillhfillhfill_ =frac=frac=frac008hfill n=7h Right is the first of the first five terms: left, frac, frac, frac, frac right.

### Analysis of the Solution

The graph of the series is seen in Figure 8. You should take note that, due to the fact that factorials grow at a very fast rate, the presence of a factorial term (or factorials) in the denominator results in the denominator becoming significantly bigger than the numerator as nincreases. This indicates that the quotient is getting smaller, and as the plot of the terms demonstrates, the terms are getting smaller and closer to zero. Image number eight (figure eight).

### Try It 9

Write the first five terms of the series specified by the explicit formula_ =frac, followed by the rest of the sequence. Solution

## Sequences as Functions – Recursive 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 recursive form.Certain sequences (not all) can be defined (expressed) in a”recursive” form. In arecursive formula,each term is defined as a function of its preceding term(s).Arecursive formuladesignates the starting term,a1, and thenthterm of the sequence,an, as an expression containing the previous term (the term before it),an-1. |

The process ofrecursioncan be thought of as climbing a ladder.To get to the third rung, you must step on the second rung.Each rung on the ladder depends upon stepping on the rung below it.You start on the first rung of the ladder.a1 From the first rung, you move to the second rung.a2a2=a1+ “step up”From the second rung, you move to the third rung.a3 a3=a2+ “step up” | |

If you are on thenthrung, you must have stepped on then -1 strung.an=an -1+ “step up” |

Notation:Recursive forms work with the term(s) immediately in front of the term being examined.The table at the right shows that there are many options as to how this relationship may be expressed innotations.A recursive formula is written withtwo parts: a statement of thefirst termalong with a statement of theformula relating successive terms.The statements below are all naming the same sequence: |

Given Term | Term in frontof Given Term |

a4 | a3 |

an | an- 1 |

an+ 1 | an |

an+ 4 | an+ 3 |

f(6) | f(5) |

f(n) | f(n- 1) |

f(n +1) | f(n) |

The following is the sequence: Find a recursive formula that works. The following is an example of 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) |

Arithmetic sequences arelinearin nature. Remember that the domain consists ofthe natural numbers, and the range consists of the terms of the sequence.It may be the case with arithmetic sequences that the graph will increase or decrease. |

In mostarithmetic sequences, a recursive formula is easier to create than an explicit formula. The common difference is usually easily seen, which is then used to quickly create the recursive formula. |

The following is the order of events: Find a formula that is recursive. In this case, the sequence is analytic (the same number, 5, is added to each term to get tothe next term).

a1= first term;a n=an -1+d | a1= the first term in the sequence a n= then thterm in the sequencea n-1= the term before then thterm n= the term numberd= the common difference. | |

first term = 10, common difference = 5recursive formula:a1 = 10;a n=a n-1+ 5 |

The following is the sequence: Find a recursive formula that works. 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. |

To explain the method of constructing a recursive formula for a geometric series, consider the following examples of writing: 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.) Using the first term as the starting point, and then declaring the formula as the common ratio multiplied by its predecessor, create a recursive formula.

a1= first term;a n=ran -1 | a1= the first term in the sequence a n= then thterm in the sequencea n-1= the term before then thtermn= the term numberr= the common ratio | |

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

Sequence:This example is neither an arithmetic sequence nor a geometric sequence in the traditional sense.

While we have seen recursive formulas forarithmetic sequences and geometric sequences, there are also recursive formsfor sequences that do not fall into either of these categories.The sequence shown in this example isa famous sequence called theFibonacci sequence. | |

Is there apattern for the Fibonacci sequence?Yes. After the first two terms, each term is thesum of the previous two terms.Is there a recursive formula forthe Fibonacci sequence?Yes.f(1) = 0;f(2) = 1;f(n)=f(n- 1) +f(n- 2) or a1= 0;a2= 1;an=an- 1+an- 2 | |

Notice that it was necessary to declare the first and second term, before stating the formula for generating the remaining terms. |

## Recursive Formula (Explained w/ 25 Step-by-Step Examples!)

Was it ever brought to your attention that a series may be defined both recursively and explicitly?

Jenn, the founder of Calcworkshop ®, has over 15 years of experience (LicensedCertified Teacher)

## What Is A Sequence

Although a sequence is strictly defined as an enumerated collection of objects, the term “sequence” may also refer to a countable structure that represents an ordered list of items or numbers. Definition And we provide a sequence either recursively or directly, depending on the situation.

## Recursive Formula Definition

So what exactly is recursion? Often referred to as aninductive definition, arecursive definition is composed of two parts: Describe recursion in further detail. Two pieces make up an arecursive definition, which is also known as aninductive definition.

## Example

For example, List Terms Recursive — An Illustration Take note that this approach is very identical to the one we use for mathematical induction! According to Math Bits, the concept of inductive proofs is analogous to that of a staircase in that the only way to reach the summit is to walk all of the steps leading up to it. In the case of recursion, the same thing happens — each step is formed from the step or steps that came before it. An Illustration of a Staircase

## Recursive Formulas For Sequences

To summarize, a recursive sequence is one in which terms are defined in terms of one or more prior terms, as well as a beginning condition, like we just discussed. The Fibonacci sequence, on the other hand, is the most well-known recursive formula. The Fibonacci sequence is composed of the following numbers: 0, 1, 2, 3, 5, 8, 13, 21,. are the digits of the number zero. Keep in mind that each number in the series is the sum of the two numbers that came before it in the sequence. For example, the number 13 is the sum of the numbers 5 and 8, which are the two phrases that came before it.

When each number in the sequence is drawn as a rectangular width, the result is a spiral formed by the sequence.

However, employing a recursive formula might be time-consuming at times, since we must constantly rely on the words that came before us in order to construct the next ones.

All this implies is that the value of each phrase in the sequence may be determined directly without having to know the value of the preceding term.

## Example

What we will note in this issue is that patterns begin to emerge when we write down the terms of our sequences in the form of terms. The qualities of recursively formed and explicitly specified sequences may be discovered in these patterns, as can be shown in the diagram below. The sequences and summations from Precalculus, such as the Arithmetic and Geometric sequences and series, are vital to recall since they will aid us in the discovery of these patterns.

Formula for the Arithmetic Sequence Formula for Geometric Sequence The Formulae for Summation Sequences Armed with these summation formulas and methodologies, we will be able to build recursive formulas and closed formulas for additional sequences that follow similar patterns and structures.

## Example

In order to find a recursive definition for the series 4,9,14,19,24,29,. let’s use our known sequences to do so. Example of Arithmetic Recursive Functions Using our previously learned summation formulas, let’s try to create a closed definition for the same series of 4, 9, 14, 19, 24, 29, and so on. Arithmetic Sequence with a Closed Form In addition, we will uncover a fantastic process for finding the sum of an Arithmetic and Geometric sequence, which will make use of Gauss’s discoveries of reverse-add and multiply-shift-subtract, respectively, to get the sum of an Arithmetic and Geometric sequence.

## Example

Consider the following sequence: 1,3,5,7,9,.,39. We want to find the sum of the numbers in this sequence. In order to complete this arithmetic series, we must first identify the closed formula for it. As a result, we must first discover the common difference, which corresponds to the amount that is being added to each phrase in order to construct the next term in the sequence, before proceeding. The quickest and most straightforward method is to remove two neighboring words. If we subtract any two neighboring words from our present example, we will note that the common difference between them is equal to two.

Calculate the number of terms in an arithmetic series by using the formula below.

To put it another way, we shall “wrap” the series around itself, as MathBitsNotebook so eloquently puts it.

## Summary

Through this video, we will show how a recursive formula calculates each term based on the value of the previous term, resulting in the sequence taking a little longer to construct than a nonrecursive formula. An explicit formula, on the other hand, explicitly calculates each term in the sequence and gets a specific term in a short amount time. When it comes to the study of counting and recurrence relations, both formulae as well as summation techniques are quite useful. With these new approaches, we will be able to not only build recursive formulae for specific sequences, but we will also be on our way to solving recurrence relations!

## Video Tutorial w/ Full LessonDetailed Examples

1 hour and 49 minutes

- Introductory Video: Recursive Formula — Sequences — Summations
- 00:00:51 000:00:51 000:00:51 000:00:51 Can you predict the pattern and decide the next phrase in the sequence? Can you guess the pattern and find the next term in the sequence? (Examples1-7)
- Members Only Access to Exclusive Content

- 00:11:37 What is the difference between a Recursive Definition and an Explicit Formula
- The sequence’s first five phrases (Examples 8–10) must be identified at the time of 00:21:43. 00:30:38Recursive formula and closed formula for Arithmetic and Geometric Sequences
- 00:40:27Recursive formula and closed formula for Arithmetic and Geometric Sequences Sequences of the shapes triangular, square, and cube, as well as exponential, factororial, and Fibonacci sequences 00:47:42 Determine the definition of each sequence using a recursive approach (Examples 11-14)
- Example 15-20 show how to use known sequences to find a closed formula
- 01:22:29 show how to use known sequences to find a closed formula When working with arithmetic sequences, use the reverse—add approach (Examples 21-22). 01:35:48 Summing Geometric Sequences Using the Multiply-Shift-Subtract Method (Examples 23-34)
- 01:44:00 01:44:00 01:44:00 01:44:00 01:44:00 Practice problems with step-by-step solutions
- Chapter tests with video solutions
- Summation and Product Notation (Examples 25a-d)

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## 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

Let’s start with a straightforward problem. We have the following numbers in our sequence: -3, 2, 7, 12,. What is the seventh and last phrase in this sequence? As we can see, the most typical difference between successive periods is five points. The fourth term equals twelve, therefore a (4) = twelve. We can continue to add terms to the list in the following order until we reach the seventh term: -3, 2, 7, 12, 17, 22, 27,. and so on. This tells us that a (7) = 27 is the answer.

## 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: