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

Contents

- 1 How do you know if a sequence is arithmetic?
- 2 Are all sequences arithmetic?
- 3 What are the 5 examples of arithmetic sequence?
- 4 Which is not arithmetic sequence?
- 5 How do you write an arithmetic sequence?
- 6 Are all sequence arithmetic sequence why?
- 7 Is Fibonacci sequence an arithmetic sequence?
- 8 Are all sequences arithmetic or geometric?
- 9 What is the arithmetic mean between 10 and 24?
- 10 What type of sequence is 80 40 20?
- 11 What is the common difference arithmetic sequence?
- 12 What are arithmetic sequences used for?
- 13 [Expert Answer] Which sequences are arithmetic? Check all that apply. –5, 5, –5, 5, –5, … 96, 48, 24, – Brainly.com
- 14 Arithmetic Sequences and Sums
- 15 Arithmetic Sequence
- 16 Advanced Topic: Summing an Arithmetic Series
- 17 Footnote: Why Does the Formula Work?
- 18 Arithmetic & Geometric Sequences
- 18.0.1 Find the common difference and the next term of the following sequence:
- 18.0.2 Find the common ratio and the seventh term of the following sequence:
- 18.0.3 Find the tenth term and then-th term of the following sequence:
- 18.0.4 Find then-th term and the first three terms of the arithmetic sequence havinga6= 5andd=
- 18.0.5 Find then-th term and the first three terms of the arithmetic sequence havinga4= 93anda8= 65.
- 18.0.6 Find then-th and the26 th terms of the geometric sequence withanda12= 160.

- 19 Arithmetic Sequences and Series
- 20 6.2: Arithmetic and Geometric Sequences
- 21 Arithmetic Sequences and Series – MathBitsNotebook(A2
- 22 Arithmetic Sequences and Series
- 23 Arithmetic Series
- 24 Arithmetic Sequences – Precalculus

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

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.

## Are all sequences arithmetic?

No. Arithmetic sequences are those sequences which can be constructed by adding a constant value to the current term to produce the next term. The counting numbers are an example of an arithmetic sequence.

## What are the 5 examples of arithmetic sequence?

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

## Which is not arithmetic sequence?

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

## How do you write an arithmetic sequence?

An arithmetic sequence is a sequence where the difference between each successive pair of terms is the same. The explicit rule to write the formula for any arithmetic sequence is this: an = a1 + d (n – 1)

## Are all sequence arithmetic sequence why?

For example, the sequence 3, 5, 7, 9 is arithmetic because the difference between consecutive terms is always two. The sequence 1, 2, 4, 8 is not arithmetic because the difference between consecutive terms is not the same.

## Is Fibonacci sequence an arithmetic sequence?

FIBONACCI SEQUENCE. If we have a sequence of numbers such as 2, 4, 6, 8, it is called an arithmetic series. He began the sequence with 0,1, and then calculated each successive number from the sum of the previous two. This sequence of numbers is called the Fibonacci Numbers or Fibonacci Sequence.

## Are all sequences arithmetic or geometric?

Not all sequences are geometric or arithmetic. For example, the Fibonacci sequence 1,1,2,3,5,8, is neither. A geometric sequence is one that has a common ratio between its elements. For example, the ratio between the first and the second term in the harmonic sequence is 121=12.

## What is the arithmetic mean between 10 and 24?

Using the average formula, get the arithmetic mean of 10 and 24. Thus, 10+24/2 =17 is the arithmetic mean.

## What type of sequence is 80 40 20?

This is a geometric sequence since there is a common ratio between each term.

## What is the common difference arithmetic sequence?

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

## What are arithmetic sequences used for?

An arithmetic sequence is a string of numbers where each number is the previous number plus a constant. This constant difference between each pair of successive numbers in our sequence is called the common difference. The general term is the formula that is used to calculate any number in an arithmetic sequence.

## [Expert Answer] Which sequences are arithmetic? Check all that apply. –5, 5, –5, 5, –5, … 96, 48, 24, – Brainly.com

1616, 32, 48, 64, 80 are the answers: 18, 5, 7, –19.5, –3216, 18, 5, 7, –19.5 The following is a step-by-step explanation: It is possible to create an arithmetic sequence from a series of integers in which the difference between successive numbers is a constant. We can see right away that the series –5, 5, –5, 5, –5,. is not an arithmetic sequence. 96, 48, 24, 12, and 6 are the numbers on the board. This is also not a mathematical sequence in any way shape or form. This is an example of a geometric sequence.

The difference between the first and second numbers is as follows: The difference between the second and third numbers is as follows: The difference between the third and fourth numbers is as follows: The difference between the 4th and 5th numbers is as follows: As a result, the difference between all of the sequential integers is a fixed value.

–1, –3, –9, –27, –81 are all negative numbers.

Essentially, it is a geometric series in which each integer is multiplied by a fixed number.

The difference between the first and second numbers is as follows: The difference between the second and third numbers is as follows: The difference between the third and fourth numbers is as follows: It is undeniable that these figures have a significant difference in value.

## Arithmetic Sequences and Sums

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

## Arithmetic Sequence

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

### Example:

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

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

### Example: (continued)

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

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

And this is what we get:

### Rule

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

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

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

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

Making use of the Arithmetic Sequencerule, we can see that_xn= a + d(n1)= 3 + 5(n1)= 3 + 3 + 5n 5 = 5n 2 xn= a + d(n1) = 3 + 3 + 3 + 5n n= 3 + 3 + 3 As a result, the ninth term is:x 9= 5 9 2= 43 Is that what you’re saying?

Take a look for yourself! Arithmetic Sequences (also known as Arithmetic Progressions (A.P.’s)) are a type of arithmetic progression.

## Advanced Topic: Summing an Arithmetic Series

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

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

The values ofa,dandnare as follows:

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

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

## Footnote: Why Does the Formula Work?

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

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

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

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

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

## Arithmetic & Geometric Sequences

The arithmetic and geometric sequences are the two most straightforward types of sequences to work with. An arithmetic sequence progresses from one term to the next by adding (or removing) the same value on each successive term. For example, the numbers 2, 5, 8, 11, 14,.are arithmetic because each step adds three; while the numbers 7, 3, –1, –5,.are arithmetic because each step subtracts four. The number that is added (or subtracted) at each stage of an arithmetic sequence is referred to as the “common difference”d because if you subtract (that is, if you determine the difference of) subsequent terms, you will always receive this common value as a result of the process.

Below In a geometric sequence, the terms are connected to one another by always multiplying (or dividing) by the same value.

Each step of a geometric sequence is represented by a number that has been multiplied (or divided), which is referred to as the “common ratio.” If you divide (that is, if you determine the ratio of) subsequent terms, you’ll always receive this common value.

#### Find the common difference and the next term of the following sequence:

3, 11, 19, 27, and 35 are the numbers. In order to get the common difference, I must remove each succeeding pair of terms from the total. There’s no point in choosing which couple I want to sit with as long as they’re right next to one other. To be thorough, I’ll go over each and every subtraction: 819 – 11 = 827 – 19 = 835 – 27 = 819 – 11 = 827 – 19 = 835 – 27 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 = 819 – 11 Due to the fact that the difference is always 8, the common difference isd=8.

By adding the common difference to the fifth phrase, I can come up with the next word: 35 plus 8 equals 43 Then here’s my response: “common difference: six-hundred-and-fortieth-term

#### Find the common ratio and the seventh term of the following sequence:

To get the common ratio, I must divide each succeeding pair of terms by the number of terms in the series. There’s no point in choosing which couple I want to sit with as long as they’re right next to one other. I’ll go over all of the divisions to be thorough: The ratio is always three, hence sor= three. As a result, I have five terms remaining; the sixth term will be the next term, and the seventh will be the term after that. The value of the seventh term will be determined by multiplying the fifth term by the common ratio two times.

When it comes to arithmetic sequences, the common difference isd, and the first terma1is commonly referred to as “a “.

As a result of this pattern, the then-th terma n will take the form: n=a+ (n– 1)d When it comes to geometric sequences, the typical ratio isr, and the first terma1 is commonly referred to as “a “.

This pattern will be followed by a phrase with the following form: a n=ar(n– 1) is equal to a n.

#### Find the tenth term and then-th term of the following sequence:

It is necessary to split a series of phrases in order to get the common ratio. As long as they’re right next to each other, it doesn’t matter which pair I choose. All of the divisions will be completed by me, just to be thorough. In all cases, the ratio is three, sor=three. Given that I was only given five semesters, the sixth semester will be immediately after that, and the seventh semester following that. I’ll multiply the fifth term by the common ratio twice more to get the value of the seventh term: 162 a6= (18)(3) = 54 a7= (54)(3) = 54 a6 = (18)(3) = 54 In such case, my response is as follows: common ratio: r= 3, seventh term: 162, and Arithmetic and geometric sequences have formulae because they are so neat and regular.

In order to proceed, the third term is as follows: a3= (a+ d) + d=a+ 2 d In the fourth term, a4= (a+ 2 d) +d=a+ 3 d, the expression is: Each stage included multiplying the common difference by a number that was one less than the index.

If we consider geometric sequences, the typical ratio isr, and the first terma1 is commonly referred to as”a.” The value ofa2is simply_a2=ar, since we derive the following term by multiplying by the common ratio_a2=ar.

This pattern will be followed by a phrase with the following structure: the value of a n=ar(n–1) the value of a Prepare for the upcoming test by memorizing thesen-th-term formulae.

#### Find then-th term and the first three terms of the arithmetic sequence havinga6= 5andd=

The n-th term in an arithmetic series has the form n=a+ (n– 1) d, which stands for n=a+ (n– 1) d. In this particular instance, that formula results in me. When I solve this formula for the value of the first term in the sequence, I obtain the resulta= Then:I have the first three terms in the series as a result of this. Because I know the value of the first term and the common difference, I can also develop the expression for the then-th term, which will be easier to remember: In such case, my response is as follows:n-th word, first three terms:

#### Find then-th term and the first three terms of the arithmetic sequence havinga4= 93anda8= 65.

Due to the fact thata4 anda8 are four places apart, I can determine from the definition of an arithmetic sequence that I can go from the fourth term to the eighth term by multiplying the common difference by four times the fourth term; in other words, the definition informs me that a8=a4 + 4 d. I can then use this information to solve for the common differenced: 65 = 93 + 4 d –28 = 4 d –7 = 65 = 93 + 4 d Also, I know that the fourth term is related to the first term by the formulaa4=a+ (4 – 1) d, so I can get the value of the first terma by using the value I just obtained ford and the value I just discovered fora: 93 =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) =a+ 3(–7) 93 plus 21 equals 114.

As soon as I know what the first term’s value is and what the value of the common difference is, I can use the plug-and-chug method to figure out what the first three terms’ values are, as well as the general form of the fourth term: The numbers are as follows: a1= 114, a2= 114– 7, a3= 107– 7, and an= 114 + (n – 1)(–7)= 114 – 7, n+ 7, and an= 121–7, respectively.

#### Find then-th and the26 th terms of the geometric sequence withanda12= 160.

Given that the two words for which they’ve provided numerical values are separated by 12 – 5 = 7 places, I know that I can go from the fifth term to the twelfth term by multiplying the fifth term by the common ratio seven times; that is, a12= (a5) (r7). I can use this to figure out what the value of the common ratior should be: I also know that the fifth component is related to the first by the formulaa5=ar4, so I can use that knowledge to solve for the value of the first term, which is as follows: Now that I know the value of the first term as well as the value of the common ratio, I can put both into the formula for the then-th term to obtain the following result: I can assess the twenty-sixth term using this formula, and it is as follows, simplified: Then here’s my response:n-th term: 2,621,440 for the 26th term Once we have mastered the art of working with sequences of arithmetic and geometric expressions, we may move on to the concerns of combining these sequences together.

## Arithmetic Sequences and Series

An arithmetic sequence is a set of integers in which the difference between the words that follow is always the same as its predecessor.

### Learning Objectives

Make a calculation for the nth term of an arithmetic sequence and then define the characteristics of arithmetic sequences.

### Key Takeaways

- When the common differenced is used, the behavior of the arithmetic sequence is determined. Arithmetic sequences may be either limited or infinite in length.

#### Key Terms

- Arithmetic sequence: An ordered list of numbers in which the difference between the subsequent terms is constant
- Endless: An ordered list of numbers in which the difference between the consecutive terms is infinite
- Infinite, unending, without beginning or end
- Limitless
- Innumerable

For example, an arithmetic progression or arithmetic sequence is a succession of integers in which the difference between the following terms is always the same as the difference between the previous terms.

A common difference of 2 may be found in the arithmetic sequence 5, 7, 9, 11, 13, cdots, which is an example of an arithmetic sequence.

- 1: The initial term in the series
- D: The difference between the common differences of consecutive terms
- A 1: a n: Then the nth term in the series.

The behavior of the arithmetic sequence is determined by the common differenced arithmetic sequence. If the common difference,d, is the following:

- Positively, the sequence will continue to develop towards infinity (+infty). If the sequence is negative, it will regress towards negative infinity (-infty)
- If it is positive, it will regress towards positive infinity (-infty).

It should be noted that the first term in the series can be thought of asa 1+0cdot d, the second term can be thought of asa 1+1cdot d, and the third term can be thought of asa 1+2cdot d, and therefore the following equation givesa n:a n In the equation a n= a 1+(n1)cdot D Of course, one may always type down each term until one has the term desired—but if one need the 50th term, this can be time-consuming and inefficient.

## 6.2: Arithmetic and Geometric Sequences

Arithmetic sequences and geometric sequences are two forms of mathematical sequences that are commonly encountered. In an arithmetic sequence, there is a constant difference between each subsequent pair of words in the sequence. There are some parallels between this and linear functions of the type (y=m x+b). Among any pair of subsequent words in a geometric series, there is a constant ratio between them. This would have the effect of a constant multiplier being applied to the data. Examples The Arithmetic Sequence is as follows: Take note that the constant difference in this case is 6.

For the n-th term, one method is to use as the coefficient the constant difference between the two terms: (a_ =6n+?).

We may state the following about the sequence: (a_ =6 n-1); (a_ =6 n-1); (a_ =6 n-1); The following is an example of a formula that you can memorize: Any integer sequence with a constant difference (d) is stated as follows: (a_ =a_ +(n-1) d) = (a_ =a_ +(n-1) d) = (a_ =a_ +(n-1) d) It’s important to note that if we use the values from our example, we receive the same result as we did before: (a_ =a_ +(n-1) d)(a_ =5, d=6)(a_ =5, d=6)(a_ =5, d=6) As a result, (a_ +(n-1) d=5+(n-1) * 6=5+6 n-6=6 n-1), or (a_ =6 n-1), or (a_ =6 n-1) A negative integer represents the constant difference when the terms of an arithmetic sequence are growing smaller as time goes on.

- (a_ =-5 n+29) (a_ =-5 n+29) (a_ =-5 n+29) Sequence of Geometric Shapes With geometric sequences, the constant multiplier remains constant throughout the whole series.
- Unless the multiplier is less than (1,) then the terms will get more tiny.
- Similarly, if the terms are becoming smaller, the multiplier would be in the denominator.
- The exercises are as follows: (a_ =frac) or (a_ =frac) or (a_ =50 *left(fracright)) and so on.
- If the problem involves arithmetic, find out what the constant difference is.

The following are examples of quads: 1) (quads), 2) (quads), and 3) (quad ) 4) (quad ) Five dots to the right of the quad (left quad, frac, frac, frac, frac, frac, frac, frac, frac, frac, frac, frac, frac, frac) 6) (quad ) 7) (quad ) 8) (quad ) 9) 9) 9) 9) 9) 9) 9) 9) (quad ) 10) 10) 10) 10) 10) (quad ) 11) (quad ) 12) (quad ) 13) (quadrilateral) (begintext end ) 15) (quad ) (No.

16) (quad )

## Arithmetic Sequences and Series – MathBitsNotebook(A2

Some sequences are composed of simply random values, while others have a definite pattern that is used to arrive at the sequence’s terms. Thearithmetic sequence(or progression),for example, is based upon the addition of a constant value to arrive at the next term in the sequence. |

When arithmetic sequences are graphed, they exhibit an alinear character (as a scatter plot). The domain of the sequence is represented by the counting numbers 1, 2, 3, 4, and so on (representing the position of each term), while the range of the sequence is represented by the actual terms of the series. While the x-axis rises by a constant value of one in the graph shown above, the y-axis increases by a constant value of three in the graph displayed above. Arithmetic Sequences are a type of mathematical sequence.

Arithmetic Sequence: | Common Difference,d: |

1, 6, 11, 16, 21, 26,. | d= 5.A 5 isaddedto each term to arrive at the next term.OR. thedifferencea2-a1= 5. |

10, 8, 6, 4, 2, 0, -2, -4,. | d= -2.A -2 isaddedto each term to arrive at the next term.OR. thedifferencea2-a1= -2. |

d= -½.A -½ isaddedto each term to arrive at the next term.OR. thedifferencea2-a1= -½. |

This is the sum of the firstnterms. |

S n=a1+(a1+ d)+(a1+2 d)+(a1+3 d)+(a1+4 d)+(a1+5 d)+(a1+6 d)+(a1+7 d)+(a1+8 d)+(a1+9 d)+(a1+(n- 1) d)+(a1+(n- 1) d)+(a When the terms of an arithmetic sequence are added together, the result is called an anarithmetic series. Formulas that are used in conjunction with arithmetic sequences and arithmetic series include:

Tofind any term of anarithmetic sequence:wherea1is the first term of the sequence, dis the common difference,nis the number of the term to find. Note:you may seea1simply referred to asa. | To find thesum of a certain number of termsof anarithmetic sequence:whereSnis the sum ofnterms (nthpartial sum),a1is the first term,a nis thenthterm. Note:(a1+a n)/2 is the mean (average) of the first and last terms. The sum can be thought of as thenumber of terms times the average of the first and last terms. This formula may also appear as |

To learn more about “How These Formulas Were Created,” please visit this page. Examine a number of examples that use arithmetic sequences and series to demonstrate their use. Read the “Answers” attentively in order to obtain ” suggestions ” on how to cope with the questions in this section.

Questions: | Answers: |

1.Find the common difference for this arithmetic sequence: 4, 15, 26, 37,. | The common difference,d,can be found bysubtracting the first term from the second term, which in this example yields 11. Checking shows that 11 is the difference between all of the terms. |

2.Find the common difference for the arithmetic sequence whose formula is:a n= 6 n+ 3. | A listing of the terms will show what is happening in the sequence (start withn= 1). 9, 15, 21, 26, 33,.The common difference is 6. |

3.Find the 10 thterm of the sequence:3, 5, 7, 9, 11,. | By observation_n= 10,a1= 3,d= 2Use the formula for thenthterm.The 10 thterm is 21. |

4.Finda7for an arithmetic sequence where_a1= 3 xandd = -x. | By observation_n= 7,a1= 3 x,d= – xYour answer will be in terms of x. |

5.Given the arithmetic sequence:f(1) = 4;f(n) =f(n- 1) + 3.Findf(5). | Don’t let the change tofunctional notationdistract you. This problem showsrecursive form:each term is defined by the term immediately in front of it.The first term is 4 and the common difference is 3. Since we only need the fifth term, we can get the answer by observation: 4, 7, 10, 13, 16f(5) = 16 |

6.Findt15for an arithmetic sequence where_t3= -4 + 5i andt6= -13 + 11 iNOTE:Using high subscript – low subscript + 1 will count the number of terms. | Notice the change of labeling fromatot.The letter used in labeling is of no importance. Let’s get a visual of this problem. Using the third terms as the “first” term, find the common difference from these known terms.Now, fromt3tot15is 13 terms. t15= -4 + 5 i+ (13-1)(-3 +2 i) = -4 + 5 i-36 +24 i = -40 + 29 i |

7.Find an explicit formula and a recursive formula for the sequence:1, 3, 5, 7, 9,. | Theexplicit formulaneeds to relate the subscript number of each term to the actual value of the term. These terms are odd numbers (a good formula pattern to remember).a n= 2 n- 1Substitutingn= 1 gives 1.Substitutionn =2 gives 3, and so on.Therecursive formula,where each term is based upon the term immediately in front of it, is easy to find since the common difference is 2.a1= 1a n= an -1+ 2. |

8.The first three terms of an arithmetic sequence are represented byx+ 5, 3 x+ 2, and 4 x+ 3 respectively. Find the numerical value of the 10 thterm of this sequence. | Represent the common difference between the terms:(3 x+ 2) – (x+ 5) = 2 x- 3 (the common difference)(4 x+ 3) – (3 x+ 2) =x+ 1 (the common difference)Since the common difference must be constant, weset these values equal and solve forx.2 x- 3 =x+ 1 x= 4 The sequence is 9, 14, 19,., common difference of 5.The 10 thterm = 9 + (10 – 1)(5) =54 |

9.Find the sum of the first 12 positive even integers.Notice how BOTH formulas work together to arriveat the answer. | The word “sum” indicates the need for the sum formula.positive even integers: 2, 4, 6, 8,. n= 12,a1= 2,d= 2We are missinga12, for the sum formula, so we use the “any term” formula to find it.Now, we use this information to find the sum: |

10.Insert 3 arithmetic means between 7 and 23.Note:In this context, anarithmetic meanis the term between any two terms of an arithmetic sequence. It is simply the average (mean) of its surrounding terms. | While there are several solution methods, we will use our arithmetic sequence formulas. Draw a picture to better understand the situation.7, _, _, _, 23This set of terms is an arithmetic sequence.We know the first term,a1, the last term,a n, but not the common difference,d.This question gives NO indicationof “sum”, so avoid that formula. Find the common difference:Now, insert the terms usingd: 7,11, 15, 18, 23 |

11.In an arithmetic sequence,a4= 19 and a7= 31. Determine the formula fora n, thenthterm of this sequence. | Visualize the problem by modeling the terms from the fourth to the seventh.19, _. _, 31Temporarily imagine that 19 is the first term.This will allow us to find the common difference.Imagined Observations_a1= 19,a 4= 31,n= 4. |

12.Find the number of terms in the sequence:7, 10, 13,., 55Note:nmustbe an integer! | By observation_a1= 7,a n= 55,d= 3. We need tofindn. This question makes No mention of “sum”, so avoid that formula.When solving forn,be sure your answer is a positiveinteger.There is no such thing as a fractional number of terms in a sequence! |

13.A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. If the theater has 20 rows of seats, how many seats are in the theater? | The seating pattern is forming an arithmetic sequence:60, 68, 76,.We need to find the “sum” of all of the seats. By observation_n= 20,a1= 60,d= 8 and we needa20for the sum.Now, use the sum formula.There are 2720 seats. |

## Arithmetic Sequences and Series

The succession of arithmetic operations There is a series of integers in which each subsequent number is equal to the sum of the preceding number and specified constants. orarithmetic progression is a type of progression in which numbers are added together. This term is used to describe a series of integers in which each subsequent number is the sum of the preceding number and a certain number of constants (e.g., 1). an=an−1+d Sequence of Arithmetic Operations Furthermore, becauseanan1=d, the constant is referred to as the common difference.

For example, the series of positive odd integers is an arithmetic sequence, consisting of the numbers 1, 3, 5, 7, 9, and so on.

This word may be constructed using the generic terman=an1+2where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, where, To formulate the following equation in general terms, given the initial terma1of an arithmetic series and its common differenced, we may write: a2=a1+da3=a2+d=(a1+d)+d=a1+2da4=a3+d=(a1+2d)+d=a1+3da5=a4+d=(a1+3d)+d=a1+4d⋮ As a result, we can see that each arithmetic sequence may be expressed as follows in terms of its initial element, common difference, and index: an=a1+(n−1)d Sequence of Arithmetic Operations In fact, every generic word that is linear defines an arithmetic sequence in its simplest definition.

### Example 1

Identify the general term of the above arithmetic sequence and use that equation to determine the series’s 100th term. For example: 7,10,13,16,19,… Solution: The first step is to determine the common difference, which is d=10 7=3. It is important to note that the difference between any two consecutive phrases is three. The series is, in fact, an arithmetic progression, with a1=7 and d=3. an=a1+(n1)d=7+(n1)3=7+3n3=3n+4 and an=a1+(n1)d=7+(n1)3=7+3n3=3n+4 and an=a1+(n1)d=3 As a result, we may express the general terman=3n+4 as an equation.

Take a moment to confirm that this equation accurately reflects the sequence you’ve been given. To determine the 100th term, use the following equation: a100=3(100)+4=304 Answer_an=3n+4;a100=304 It is possible that the common difference of an arithmetic series be negative.

### Example 2

Identify the general term of the given arithmetic sequence and use it to calculate the 75th term of the sequence: 6,4,2,0,−2,… Solution: Make a start by determining the common difference, d = 4 6=2. Next, find the formula for the general term, wherea1=6andd=2 are the variables. an=a1+(n−1)d=6+(n−1)⋅(−2)=6−2n+2=8−2n As a result, an=8nand the 75thterm can be calculated as follows: an=8nand the 75thterm a75=8−2(75)=8−150=−142 Answer_an=8−2n;a100=−142 The terms in an arithmetic sequence that occur between two given terms are referred to as arithmetic means.

### Example 3

Find all of the words that fall between a1=8 and a7=10. in the context of an arithmetic series Or, to put it another way, locate all of the arithmetic means between the 1st and 7th terms. Solution: Begin by identifying the points of commonality. In this situation, we are provided with the first and seventh terms, respectively: an=a1+(n−1) d Make use of n=7.a7=a1+(71)da7=a1+6da7=a1+6d Substitutea1=−8anda7=10 into the preceding equation, and then solve for the common differenced result. 10=−8+6d18=6d3=d Following that, utilize the first terma1=8.

a1=3(1)−11=3−11=−8a2=3 (2)−11=6−11=−5a3=3 (3)−11=9−11=−2a4=3 (4)−11=12−11=1a5=3 (5)−11=15−11=4a6=3 (6)−11=18−11=7} In arithmetic, a7=3(7)11=21=10 means a7=3(7)11=10 Answer: 5, 2, 1, 4, 7, and 8.

### Example 4

Find the general term of an arithmetic series with a3=1 and a10=48 as the first and last terms. Solution: We’ll need a1 and d in order to come up with a formula for the general term. Using the information provided, it is possible to construct a linear system using these variables as variables. andan=a1+(n−1) d:{a3=a1+(3−1)da10=a1+(10−1)d⇒ {−1=a1+2d48=a1+9d Make use of a3=1. Make use of a10=48. Multiplying the first equation by one and adding the result to the second equation will eliminate a1.

an=a1+(n−1)d=−15+(n−1)⋅7=−15+7n−7=−22+7n Answer_an=7n−22 Take a look at this!

For example: 32,2,52,3,72,… Answer_an=12n+1;a100=51

## Arithmetic Series

Series of mathematical operations When an arithmetic sequence is added together, the result is called the sum of its terms (or the sum of its terms and numbers). Consider the following sequence: S5=n=15(2n1)=++++= 1+3+5+7+9+25=25, where S5=n=15(2n1)=++++ = 1+3+5+7+9=25, where S5=n=15(2n1)=++++= 1+3+5+7+9 = 25. Adding 5 positive odd numbers together, like we have done previously, is manageable and straightforward. Consider, on the other hand, adding the first 100 positive odd numbers. This would be quite time-consuming.

When we write this series in reverse, we get Sn=an+(and)+(an2d)+.+a1 as a result.

2.:Sn=n(a1+an) 2 Calculate the sum of the first 100 terms of the sequence defined byan=2n1 by using this formula. There are two variables, a1 and a100. The sum of the two variables, S100, is 100 (1 + 100)2 = 100(1 + 199)2.

### Example 5

The sum of the first 50 terms of the following sequence: 4, 9, 14, 19, 24,. is to be found. The solution is to determine whether or not there is a common difference between the concepts that have been provided. d=9−4=5 It is important to note that the difference between any two consecutive phrases is 5. The series is, in fact, an arithmetic progression, and we may writean=a1+(n1)d=4+(n1)5=4+5n5=5n1 as an anagram of the sequence. As a result, the broad phrase isan=5n1 is used. For this sequence, we need the 1st and 50th terms to compute the 50thpartial sum of the series: a1=4a50=5(50)−1=249 Then, using the formula, find the partial sum of the given arithmetic sequence that is 50th in length.

### Example 6

Evaluate:Σn=135(10−4n). This problem asks us to find the sum of the first 35 terms of an arithmetic series with a general terman=104n. The solution is as follows: This may be used to determine the 1 stand for the 35th period. a1=10−4(1)=6a35=10−4(35)=−130 Then, using the formula, find out what the 35th partial sum will be. Sn=n(a1+an)2S35=35⋅(a1+a35)2=352=35(−124)2=−2,170 2,170 is the answer.

### Example 7

Evaluate:Σn=135(10−4n). Solved:In this scenario, we are requested to find the sum of the first 35 terms of an arithmetic series having a general terman of 104n. The 1 stand the 35thterm is determined by using this. a1=10−4(1)=6a35=10−4(35)=−130 Then, using the formula, find out what the 35th partial sum equals to. Sn=n(a1+an)2S35=35⋅(a1+a35)2=352=35(−124)2=−2,170 2,170 is the correct response.

### Key Takeaways

- When the difference between successive terms is constant, a series is called an arithmetic sequence. According to the following formula, the general term of an arithmetic series may be represented as the sum of its initial term, common differenced term, and indexnumber, as follows: an=a1+(n−1)d
- An arithmetic series is the sum of the terms of an arithmetic sequence
- An arithmetic sequence is the sum of the terms of an arithmetic series
- As a result, the partial sum of an arithmetic series may be computed using the first and final terms in the following manner: Sn=n(a1+an)2

### Topic Exercises

- Given the first term and common difference of an arithmetic series, write the first five terms of the sequence. Calculate the general term for the following numbers: a1=5
- D=3
- A1=12
- D=2
- A1=15
- D=5
- A1=7
- D=4
- D=1
- A1=23
- D=13
- A 1=1
- D=12
- A1=54
- D=14
- A1=1.8
- D=0.6
- A1=4.3
- D=2.1

- Find a formula for the general term based on the arithmetic sequence and apply it to get the 100 th term based on the series. 0.8, 2, 3.2, 4.4, 5.6,.
- 4.4, 7.5, 13.7, 16.8,.
- 3, 8, 13, 18, 23,.
- 3, 7, 11, 15, 19,.
- 6, 14, 22, 30, 38,.
- 5, 10, 15, 20, 25,.
- 2, 4, 6, 8, 10,.
- 12,52,92,132,.
- 13, 23, 53,83,.
- 14,12,54,2,114,. Find the positive odd integer that is 50th
- Find the positive even integer that is 50th
- Find the 40 th term in the sequence that consists of every other positive odd integer in the following format: 1, 5, 9, 13,.
- Find the 40th term in the sequence that consists of every other positive even integer: 1, 5, 9, 13,.
- Find the 40th term in the sequence that consists of every other positive even integer: 2, 6, 10, 14,.
- 2, 6, 10, 14,. What number is the term 355 in the arithmetic sequence 15, 5, 5, 15, 25,.
- What number is the phrase 172 in the arithmetic sequence 4, 4, 12, 20, 28,.
- What number is the term 355 in the arithmetic sequence 15, 5, 5, 15, 25,.
- Find an equation that yields the general term in terms of a1 and the common differenced given the arithmetic sequence described by the recurrence relationan=an1+5wherea1=2 andn1 and the common differenced
- Find an equation that yields the general term in terms ofa1and the common differenced, given the arithmetic sequence described by the recurrence relationan=an1-9wherea1=4 andn1
- This is the problem.

- Calculate a formula for the general term based on the terms of an arithmetic sequence: a1=6anda7=42
- A1=12anda12=6
- A1=19anda26=56
- A1=9anda31=141
- A1=16anda10=376
- A1=54anda11=654
- A3=6anda26=40
- A3=16andananda15=

- Find all possible arithmetic means between the given terms: a1=3anda6=17
- A1=5anda5=7
- A2=4anda8=7
- A5=12anda9=72
- A5=15anda7=21
- A6=4anda11=1
- A7=4anda11=1

### Part B: Arithmetic Series

- Make a calculation for the provided total based on the formula for the general term an=3n+5
- S100
- An=5n11
- An=12n
- S70
- An=132n
- S120
- An=12n34
- S20
- An=n35
- S150
- An=455n
- S65
- An=2n48
- S95
- An=4.41.6n
- S75
- An=6.5n3.3
- S67
- An=3n+5

- Consider the following values: n=1160(3n)
- N=1121(2n)
- N=1250(4n3)
- N=1120(2n+12)
- N=170(198n)
- N=1220(5n)
- N=160(5212n)
- N=151(38n+14)
- N=1120(1.5n2.6)
- N=1175(0.2n1.6)
- The total of all 200 positive integers is found by counting them up. To solve this problem, find the sum of the first 400 positive integers.

- The generic term for a sequence of positive odd integers is denoted byan=2n1 and is defined as follows: Furthermore, the generic phrase for a sequence of positive even integers is denoted by the number an=2n. Look for the following: The sum of the first 50 positive odd integers
- The sum of the first 200 positive odd integers
- The sum of the first 50 positive even integers
- The sum of the first 200 positive even integers
- The sum of the first 100 positive even integers
- The sum of the firstk positive odd integers
- The sum of the firstk positive odd integers the sum of the firstk positive even integers
- The sum of the firstk positive odd integers
- There are eight seats in the front row of a tiny theater, which is the standard configuration. Following that, each row contains three additional seats than the one before it. How many total seats are there in the theater if there are 12 rows of seats? In an outdoor amphitheater, the first row of seating comprises 42 seats, the second row contains 44 seats, the third row contains 46 seats, and so on and so forth. When there are 22 rows, how many people can fit in the theater’s entire seating capacity? The number of bricks in a triangle stack are as follows: 37 bricks on the bottom row, 34 bricks on the second row and so on, ending with one brick on the top row. What is the total number of bricks in the stack
- Each succeeding row of a triangle stack of bricks contains one fewer brick, until there is just one brick remaining on the top of the stack. Given a total of 210 bricks in the stack, how many rows does the stack have? A wage contract with a 10-year term pays $65,000 in the first year, with a $3,200 raise for each consecutive year after. Calculate the entire salary obligation over a ten-year period (see Figure 1). In accordance with the hour, a clock tower knocks its bell a specified number of times. The clock strikes once at one o’clock, twice at two o’clock, and so on until twelve o’clock. A day’s worth of time is represented by the number of times the clock tower’s bell rings.

### Part C: Discussion Board

- Is the Fibonacci sequence an arithmetic series or a geometric sequence? How to explain: Using the formula for the then th partial sum of an arithmetic sequenceSn=n(a1+an)2and the formula for the general terman=a1+(n1)dto derive a new formula for the then th partial sum of an arithmetic sequenceSn=n2, we can derive the formula for the then th partial sum of an arithmetic sequenceSn=n2. How would this formula be beneficial in the given situation? Explain with the use of an example of your own creation
- Discuss strategies for computing sums in situations when the index does not begin with one. For example, n=1535(3n+4)=1,659
- N=1535(3n+4)=1,659
- Carl Friedrich Gauss is the subject of a well-known tale about his misbehaving in school. As a punishment, his instructor assigned him the chore of adding the first 100 integers to his list of disciplinary actions. According to folklore, young Gauss replied accurately within seconds of being asked. The question is, what is the solution, and how do you believe he was able to come up with the figure so quickly?

### Answers

- 5, 8, 11, 14, 17
- An=3n+2
- 15, 10, 5, 0, 0
- An=205n
- 12,32,52,72,92
- An=n12
- 1,12, 0,12, 1
- An=3212n
- 1.8, 2.4, 3, 3.6, 4.2
- An=0.6n+1.2
- An=6n3
- A100=597
- An=14n
- A100=399
- An=5n
- A100=500
- An=2n32

- 2,450, 90, 7,800, 4,230, 38,640, 124,750, 18,550, 765, 10,000, 20,100, 2,500, 2,550, K2, 294 seats, 247 bricks, $794,000, and

## 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.

### 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? If this is the case, identify the common difference.

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

- 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 frequently 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 main distinction is whether or not the provided sequence is arithmetic.
- 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

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

For an Arithmetic Sequence, a Recursive Formula can be used. For an arithmetic series with common difference, the recursive formula is as follows: to write the recursive formula for an arithmetic series given an input sequence

- 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

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

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

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

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

- 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 often encountered difference is ten.

To simplify the formula, substitute the common difference and the first term in the series into it. Analysis It can be shown in (Figure) that the slope of this sequence is 10 and that the vertical intercept is 10. For the arithmetic series that follows, provide an explicit formula for it.

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

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

The most noticeable change is.

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

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

#### Solving Application Problems with Arithmetic Sequences

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

- 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 an initial term of 1 and a common difference of 2 can 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 sequence, we get the following results: By subtracting, we can find out how many years have passed since we were five. We’re looking 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 formula. What will the child’s allowance be when he or she reaches the age of sixteen? 23 hours per 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.

#### 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 To go to the TBLSET, press the button. To go to the TABLE, press the button. 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.

Make careful to modify the WINDOW settings as necessary.

#### 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