What Does A Mean In Arithmetic Sequence? (TOP 5 Tips)





an+1= an + d, d — . : , d.

Contents

What is the an in arithmetic sequence?

The number a is the first term, and d is the common difference of the. sequence. The nth term of an arithmetic sequence is given by. an = a + (n – 1)d. The number d is called the common difference because any two consecutive terms of an.

What does a1 mean in arithmetic sequences?

“The nth term of an arithmetic sequence is an = a1 + (n – 1) d, where a1 is the first term and d is the common difference.”

What is Z in arithmetic sequence?

Jun 24, 2018. Assuming r is the constant difference between two consecutive terms, you express z=y+r in terms of y and z=x+2r in terms of x.

What is nth term?

The nth term is a formula that enables us to find any term in a sequence. The ‘n’ stands for the term number. To find the 10th term we would follow the formula for the sequence but substitute 10 instead of ‘n’; to find the 50th term we would substitute 50 instead of n.

What is the meaning of terms in mathematics?

A term is a single mathematical expression. It may be a single number (positive or negative), a single variable ( a letter ), several variables multiplied but never added or subtracted. The number in front of a term is called a coefficient. Examples of single terms: 3x is a single term. The “3” is a coefficient.

What does a3 mean in math?

a3 = 20. 6, 12, 20. Arithmetic Sequences. Definition: An arithmetic sequence is a sequence in which each term, after the first, is the sum of the preceding term and a common difference. Example: In the sequence 2, 5, 8, 11, 14,

What is the value of n in arithmetic sequence?

All you need to do is plug the given values into the formula tn = a + (n – 1) d and solve for n, which is the number of terms. Note that tn is the last number in the sequence, a is the first term in the sequence, and d is the common difference.

What is an a1 n 1 d?

If a1 is the first term of an arithmetic sequence and d is the common difference, then the formula for finding the nth term of the sequence is an = a1 + (n – 1)d.

What is the difference between an and N in arithmetic progression?

N stands for the number of terms while An stands for the nth term it ISNT the number of terms. Don’t get confused. Cheers!

What is the meaning of geometric sequence?

A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant. We can write a formula for the n th term of a geometric sequence in the form.

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:

Intro to arithmetic sequences

What I want to accomplish in this video is introduce us to a very typical class of sequences that we will encounter in the future. This is an example of arithmetic sequences. Furthermore, they are typically rather straightforward to identify. They are sequences in which each term is a defined number of times greater than the term before it, as seen in the diagram. So my aim is to figure out which of these sequences are arithmetic sequences in order to do this. In order to give us some practice with the sequence notation, I’d want to define them either as explicit functions of the phrase you’re looking for, the index you’re looking at, or as recursive definitions, just so we can get some practice with it as well.

  1. Let’s have a look at this first one, which is located over here.
  2. Then, in order to move from negative 3 to negative 1, you must multiply by 2.
  3. As a result, it is evident that this is an arithmetic series.
  4. And there are a number of other ways in which we may define the sequence.
  5. Furthermore, you are not need to utilize the letter k.
  6. From n = 1 to infinity with—and there are two ways to define it—we have a problem.
  7. We might thus write a sub n equals whatever the first word is to describe it explicitly if we wanted to be specific about what we meant.

In this case, it’s equal to negative 5 plus—we’ll add 2 one less time than the term we’re now at.

For the third term, we multiply by 2 times more.

As a result, we’re planning to add 2.

It follows that the following is an explicit definition of this arithmetic series Alternatively, if I wanted to state it in a recursive manner, I might say that a sub 1 equals negative 5.

Each phrase is equivalent to the preceding term—not 3-plus-2, but 3-plus-3.

In other words, each of these options is a perfectly legal approach to define the arithmetic sequence that we have here.

Take a look at the following sequence.

We’re starting from the beginning.

107 to 114, we’re going to add 7.

As a result, this is a valid arithmetic sequence.

For example, if we want to define it specifically, we could write that this is the sequence a sub n, n running from 1 to infinity of- and we could simply say that a sub n is equal to 100 plus we’re adding 7 every time, if we don’t want to express it properly.

In the third term, we multiply by 7 times.

It is so explicitly defined here, but we could instead define it recursively in the following way: Simply said, this is one definition where we express it like this, or we could write a subn, which would be from n = 1 to infinity, or something similar.

I could also claim that a sub 1 is equal to 100 if I wanted to define it in a recursive manner.

And with that, we’re done.

Assuming you’re looking for a generalizable approach to identify or describe an arithmetic sequence, you might state that an arithmetic sequence is going to have the form a sub n- if we’re talking about an infinite series- from n equals 1 to infinity.

It would be some constant plus some number that you are incrementing- alternatively, I assume, this might be a negative number or decrementing by- times n minus 1.

As a result, this is one method of defining an arithmetic sequence.

In this situation, the value of d is 7.

And in this situation, k is a negative 5, and in this case, k is a hundred (k).

In the case of n larger than or equal to 2, the provided term is equal to the preceding term plus d.

This is the recursive approach of putting things into words.

Now, the remaining question I have is whether or not this series over here is an arithmetic sequence.

We’ll start with the number one.

Then we add three more.

Now we’re going to add a fourth.

So, first and foremost, this is not arithmetic in any way.

But, given that we’re attempting to define our sequences, how would we go about doing so?

Consequently, we may argue that this is equivalent to a sub n, where n begins at 1 and continues to infinity, with—call let’s it our base case—a sub 1 equal to 1.

As a result, a sub 2 equals the previous term plus 2, a sub 3 equals the previous phrase plus 3, and a sub 4 equals the previous term plus 4.

Consequently, while this appears to be a close match, keep in mind that the quantity that we’re adding varies depending on our index.

Thus, when n is higher than or equal to 2, this is the case.

In the case of an arithmetic series, we’re always adding the same amount, regardless of where we are in the sequence. We’re going to add the index itself here. As a result, this is not an arithmetic sequence, but it is an intriguing one anyway.

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.

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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:

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.

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 determine the 75th term of the series: 6,4,2,0,−2,… Solution: Make a start by determining the common difference, d = 4 6=2. Next, determine 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 may be determined 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 provided 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! Identify the general term of the above arithmetic sequence and use that equation to determine the series’s 100th term. 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.

The sum of the two variables, S100, is 100 (1 + 100)2 = 100(1 + 199)2.

Example 5

Sequences of numbers in an algebraic notation When an arithmetic sequence is added together, the result is called the sum of its terms (or the sum of its terms and terms and terms). Consider the following sequence: S5=n=15(2n1)=++++= 1+3+5+7+9+25=25, where the total of the first 5 terms is defined byan=2n1. Making a sum of 5 positive odd integers is manageable; we have done it above. Consider, on the other hand, adding the first 100 positive odd numbers as a starting point. This would be an extremely time-consuming undertaking.

Sn=an+(and)+(an2d)+.+a1 is the result of reversing this sequence.

In arithmetic, the sum of the firstn terms of an arithmetic sequence is given by the formula_Sn=n(a1+an).

S100=100(a1+a100)2=100(1+199)2=10,000 since a1=1 and a100=199.

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

In an outdoor amphitheater, the first row of seating comprises 26 seats, the second row contains 28 seats, the third row contains 30 seats, and so on and so forth. Is there a maximum capacity for seating in the theater if there are 18 rows of seats? The Roman Theater (Fig. 9.2) (Wikipedia) Solution: To begin, discover a formula that may be used to calculate the number of seats in each given row. In this case, the number of seats in each row is organized into a sequence: 26,28,30,… It is important to note that the difference between any two consecutive words is 2.

  1. where a1=26 and d=2.
  2. As a result, the number of seats in each row may be calculated using the formulaan=2n+24.
  3. In order to do this, we require the following 18 thterms: a1=26a18=2(18)+24=60 This may be used to calculate the 18th partial sum, which is calculated as follows: Sn=n(a1+an)2S18=18⋅(a1+a18)2=18(26+60) 2=9(86)=774 There are a total of 774 seats available.
  4. Calculate the sum of the first 60 terms of the following sequence of numbers: 5, 0, 5, 10, 15,.
  5. Answer:S60=−8,550

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

  1. 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
  2. D=3
  3. A1=12
  4. D=2
  5. A1=15
  6. D=5
  7. A1=7
  8. D=4
  9. D=1
  10. A1=23
  11. D=13
  12. A 1=1
  13. D=12
  14. A1=54
  15. D=14
  16. A1=1.8
  17. D=0.6
  18. A1=4.3
  19. D=2.1
  1. Given the first term and the common difference of an arithmetic series, write the first five terms of the sequence in the appropriate format. Determine its generic expression using these numbers: a1=5
  2. D=3
  3. A1=12
  4. D=2
  5. A1=15
  6. D=5
  7. A1=7
  8. D=4
  9. D=1
  10. A1=23
  11. D=13
  12. A1=1
  13. D=12
  14. A1=54
  15. D=14
  16. A1=1.8
  17. D=0.6
  18. A1=4.3
  19. D=2.1
  1. Calculate a formula for the general term based on the terms of an arithmetic sequence: a1=6anda7=42
  2. A1=12anda12=6
  3. A1=19anda26=56
  4. A1=9anda31=141
  5. A1=16anda10=376
  6. A1=54anda11=654
  7. A3=6anda26=40
  8. A3=16andananda15=
  1. Find all possible arithmetic means between the given terms: a1=3anda6=17
  2. A1=5anda5=7
  3. A2=4anda8=7
  4. A5=12anda9=72
  5. A5=15anda7=21
  6. A6=4anda11=1
  7. A7=4anda11=1

Part B: Arithmetic Series

  1. Make a calculation for the provided total based on the formula for the general term an=3n+5
  2. S100
  3. An=5n11
  4. An=12n
  5. S70
  6. An=132n
  7. S120
  8. An=12n34
  9. S20
  10. An=n35
  11. S150
  12. An=455n
  13. S65
  14. An=2n48
  15. S95
  16. An=4.41.6n
  17. S75
  18. An=6.5n3.3
  19. S67
  20. An=3n+5
  1. Consider the following values: n=1160(3n)
  2. N=1121(2n)
  3. N=1250(4n3)
  4. N=1120(2n+12)
  5. N=170(198n)
  6. N=1220(5n)
  7. N=160(5212n)
  8. N=151(38n+14)
  9. N=1120(1.5n2.6)
  10. N=1175(0.2n1.6)
  11. 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.
  1. 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
  2. The sum of the first 200 positive odd integers
  3. The sum of the first 50 positive even integers
  4. The sum of the first 200 positive even integers
  5. The sum of the first 100 positive even integers
  6. The sum of the firstk positive odd integers
  7. The sum of the firstk positive odd integers the sum of the firstk positive even integers
  8. The sum of the firstk positive odd integers
  9. 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
  10. 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

  1. 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
  2. Discuss strategies for computing sums in situations when the index does not begin with one. For example, n=1535(3n+4)=1,659
  3. N=1535(3n+4)=1,659
  4. 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

  1. Is the Fibonacci sequence an arithmetic series, or is it a mathematical 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 certain situations? Make a personal example to illustrate your point
  2. Discuss strategies for computing sums in situations when the index does not begin at one (1). n=1535(3n+4)=1,659 is an example of the number n=1535(3n+4)=1,659 Carl Friedrich Gauss was once accused of misbehaving at school, according to a well-known legend. As a punishment, his instructor assigned him the chore of adding the first 100 integers to his list of disciplinary measures. Apparently, Gauss responded accurately within seconds, according to mythology. In what way do you believe he was able to come up with the solution so rapidly, and how do you think he did it?
  1. 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
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Arithmetic Sequences

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

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

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

Arithmetic progression – Wikipedia

In mathematics, an arithmetic sequence is a succession of integers in which the value of each number grows or decreases by a fixed amount each term. Thenthterm in an arithmetic series may be represented by the formula forman=dn+c, where d is the common difference. As soon as you’ve determined the common difference, you can calculate the value ofcby substituting 1 fornand the first term in the series fora1 into the equation. 1, 5, 9, 13, 17, 21, 25, is an arithmetic series having a common difference of 4, as seen in Example 1: 1, 5, 9, 13, 17, 21, 25,.

  1. So, in order to get a formula for the thenthterm, we replace n = 1, a1 = 1, andd = 4 in an= dn + c to findc.
  2. As an example, the arithmetic sequence 12-9-6-3-0-3-6) is an arithmetic series having a common difference of three.
  3. It is important to note that, because the series is decreasing, the common difference is a negative number.
  4. Thenthterm is found by substitutingn=1, a1=12, and d=3inan=dn+c to get the formula n=1, a1=12, and d=3inan=dn+c to get the formula c.12 = 3(1) plus cc = 15.
  5. As an illustration, consider the third case.
  6. In this case, the first difference is 1, and the second is 2, but the third is also 2.
  7. Geometric sequences are another term for this.

Sum

2 + 5 + 8 + 11 + 14 = 40
14 + 11 + 8 + 5 + 2 = 40

16 + 16 + 16 + 16 + 16 = 80

Calculation of the total amount 2 + 5 + 8 + 11 + 14 = 2 + 5 + 8 + 11 + 14 When the sequence is reversed and added to itself term by term, the resultant sequence has a single repeating value equal to the sum of the first and last numbers (2 + 14 = 16), which is the sum of the first and final numbers in the series. As a result, 16 + 5 = 80 is double the total. When all the elements of a finite arithmetic progression are added together, the result is known as anarithmetic series. Consider the following sum, for example: To rapidly calculate this total, begin by multiplying the number of words being added (in this case 5), multiplying by the sum of the first and last numbers in the progression (in this case 2 + 14 = 16), then dividing the result by two: In the example above, this results in the following equation: This formula is applicable to any real numbers and.

Derivation

An animated demonstration of the formula that yields the sum of the first two numbers, 1+2+.+n. Start by stating the arithmetic series in two alternative ways, as shown above, in order to obtain the formula. When both sides of the two equations are added together, all expressions involvingdcancel are eliminated: The following is a frequent version of the equation where both sides are divided by two: After re-inserting the replacement, the following variant form is produced: Additionally, the mean value of the series may be computed using the following formula: The formula is extremely close to the mean of an adiscrete uniform distribution in terms of its mathematical structure.

Product

When the members of a finite arithmetic progression with a beginning elementa1, common differencesd, andnelements in total are multiplied together, the product is specified by a closed equation where indicates the Gamma function. When the value is negative or 0, the formula is invalid. This is a generalization of the fact that the product of the progressionis provided by the factorialand that the productforpositive integersandis supplied by the factorial.

Derivation

Where represents the factorial ascension. According to the recurrence formula, which is applicable for complex numbers0 “In order to have a positive complex number and an integer that is greater than or equal to 1, we need to divide by two. As a result, if0 “as well as a concluding note

Examples

Exemple No. 1 If we look at an example, up to the 50th term of the arithmetic progression is equal to the product of all the terms. The product of the first ten odd numbers is provided by the number = 654,729,075 in Example 2.

Standard deviation

In any mathematical progression, the standard deviation may be determined aswhere is the number of terms in the progression and is the common difference between terms. The standard deviation of an adiscrete uniform distribution is quite close to the standard deviation of this formula.

Intersections

The intersection of any two doubly infinite arithmetic progressions is either empty or another arithmetic progression, which may be obtained using the Chinese remainder theorem. The intersection of any two doubly infinite arithmetic progressions is either empty or another arithmetic progression. Whenever a pair of progressions in a family of doubly infinite arithmetic progressions has a non-empty intersection, there exists a number that is common to all of them; in other words, infinite arithmetic progressions form a Helly family.

History

This method was invented by a young Carl Friedrich Gaussin primary school student who, according to a story of uncertain reliability, multiplied n/2 pairs of numbers in the sum of the integers from 1 through 100 by the values of each pairn+ 1. This method is used to compute the sum of the integers from 1 through 100. However, regardless of whether or not this narrative is true, Gauss was not the first to discover this formula, and some believe that its origins may be traced back to the Pythagoreans in the 5th century BC.

See also

  • Geometric progression
  • Harmonic progression
  • Arithmetic progression
  • Number with three sides
  • Triangular number
  • Sequence of arithmetic and geometry operations
  • Inequality between the arithmetic and geometric means
  • In mathematical progression, primes are used. Equation of difference in a linear form
  • A generalized arithmetic progression is a set of integers that is formed in the same way that an arithmetic progression is, but with the addition of the ability to have numerous different differences
  • Heronian triangles having sides that increase in size as the number of sides increases
  • Mathematical problems that include arithmetic progressions
  • Utonality

References

  1. Duchet, Pierre (1995), “Hypergraphs,” in Graham, R. L., Grötschel, M., and Lovász, L. (eds. ), Handbook of combinatorics, Vol. 1, 2, Amsterdam: Elsevier, pp. 381–432, MR1373663. Duchet, Pierre (1995), “Hypergraphs,” in Graham, R. L., Grötschel, M., and Particularly noteworthy are Section 2.5, “Helly Property,” pages 393–394
  2. And Hayes, Brian (2006). “Gauss’s Day of Reckoning,” as the saying goes. Journal of the American Scientist, 94(3), 200, doi:10.1511/2006.59.200 The original version of this article was published on January 12, 2012. retrieved on October 16, 2020
  3. Retrieved on October 16, 2020
  4. “The Unknown Heritage”: a trace of a long-forgotten center of mathematical expertise,” J. Hyrup, et al. The American Journal of Physics 62, 613–654 (2008)
  5. Tropfke, Johannes, et al (1924). Geometrie analytisch (analytical geometry) pp. 3–15. ISBN 978-3-11-108062-8
  6. Tropfke, Johannes. Walter de Gruyter. pp. 3–15. ISBN 978-3-11-108062-8
  7. (1979). Arithmetik and Algebra are two of the most important subjects in mathematics. pp. 344–354, ISBN 978-3-11-004893-3
  8. Problems to Sharpen the Young,’ Walter de Gruyter, pp. 344–354, ISBN 978-3-11-004893-3
  9. The Mathematical Gazette, volume 76, number 475 (March 1992), pages 102–126
  10. Ross, H.E.Knott, B.I. (2019) Dicuil (9th century) on triangle and square numbers, British Journal for the History of Mathematics, volume 34, number 2, pages 79–94
  11. Laurence E. Sigler is the translator for this work (2002). The Liber Abaci of Fibonacci. Springer-Verlag, Berlin, Germany, pp.259–260, ISBN 0-387-95419-8
  12. Victor J. Katz is the editor of this work (2016). The Mathematics of Medieval Europe and North Africa: A Sourcebook is a reference work on medieval mathematics. 74.23 A Mediaeval Derivation of the Sum of an Arithmetic Progression. Princeton, NJ: Princeton University Press, 1990, pp. 91, 257. ISBN 9780691156859
  13. Stern, M. (1990). 74.23 A Mediaeval Derivation of the Sum of an Arithmetic Progression. Princeton, NJ: Princeton University Press, 1990, pp. 91, 257. ISBN 9780691156859
  14. Stern, M. Journal of the American Mathematical Society, vol. 74, no. 468, pp. 157-159. doi:10.2307/3619368.

External links

  • Weisstein, Eric W., “Arithmetic series,” in Encyclopedia of Mathematics, EMS Press, 2001
  • “Arithmetic progression,” in Encyclopedia of Mathematics, EMS Press, 2001. MathWorld
  • Weisstein, Eric W. “Arithmetic series.” MathWorld
  • Weisstein, Eric W. “Arithmetic series.”

What is an Arithmetic Sequence?

Sequences of numbers are useful in algebra because they allow you to see what occurs when something keeps becoming larger or smaller over time. The common difference, which is the difference between one number and the next number in the sequence, is the defining characteristic of an arithmetic sequence.

This difference is a constant value in arithmetic sequences, and it can be either positive or negative in nature. Consequently, an arithmetic sequence continues to grow or shrink by a defined amount each time a new number is added to the list of numbers that make up the sequence is added to it.

TL;DR (Too Long; Didn’t Read)

As defined by the Common Difference formula, an arithmetic sequence is a list of integers in which consecutive entries differ by the same amount, called the common difference. Whenever the common difference is positive, the sequence continues to grow by a predetermined amount, and when it is negative, the series begins to shrink. The geometric series, in which terms differ by a common factor, and the Fibonacci sequence, in which each number is the sum of the two numbers before it, are two more typical sequences that might be encountered.

How an Arithmetic Sequence Works

There are three elements that form an arithmetic series: a starting number, a common difference, and the number of words in the sequence. For example, the first twelve terms of an arithmetic series with a common difference of three and five terms are 12, 15, 18, 21, and 24. A declining series starting with the number 3 has a common difference of 2 and six phrases, and it is an example of a decreasing sequence. This series is composed of the numbers 3, 1, 1, 3, 5, and 7. There is also the possibility of an unlimited number of terms in arithmetic sequences.

Arithmetic Mean

A matching series to an arithmetic sequence is a series that sums all of the terms in the sequence. When the terms are put together and the total is divided by the total number of terms, the result is the arithmetic mean or the mean of the sum of the terms. The arithmetic mean may be calculated using the formula text = frac n text. The observation that when the first and last terms of an arithmetic sequence are added, the total is the same as when the second and next to last terms are added, or when the third and third to last terms are added, provides a simple method of computing the mean of an arithmetic series.

The mean of an arithmetic sequence is calculated by dividing the total by the number of terms in the sequence; hence, the mean of an arithmetic sequence is half the sum of the first and final terms.

Instead, by restricting the total to a specific number of items, it is possible to find the mean of a partial sum.

Other Types of Sequences

Observations from experiments or measurements of natural occurrences are frequently used to create numerical sequences. Such sequences can be made up of random numbers, although they are more typically made up of arithmetic or other ordered lists of numbers than random numbers. Geometric sequences, as opposed to arithmetic sequences, vary in that they share a common component rather than a common difference in their composition. To avoid the repetition of the same number being added or deleted for each new phrase, a number is multiplied or divided for each new term that is added.

Other sequences are governed by whole distinct sets of laws.

The numbers are as follows: 1, 1, 2, 3, 5, 8, and so on.

Arithmetic sequences are straightforward, yet they have a variety of practical applications. As long as the starting point is known and the common difference can be identified, the value of the series at a specified point in the future may be estimated, as well as the average value.

Arithmetic Sequences and Series

HomeLessonsArithmetic Sequences and Series Updated July 16th, 2020
Introduction
Sequences of numbers that follow a pattern of adding a fixed number from one term to the next are called arithmetic sequences. The following sequences are arithmetic sequences:Sequence A:5, 8, 11, 14, 17,.Sequence B:26, 31, 36, 41, 46,.Sequence C:20, 18, 16, 14, 12,.Forsequence A, if we add 3 to the first number we will get the second number.This works for any pair of consecutive numbers.The second number plus 3 is the third number: 8 + 3 = 11, and so on.Forsequence B, if we add 5 to the first number we will get the second number.This also works for any pair of consecutive numbers.The third number plus 5 is the fourth number: 36 + 5 = 41, which will work throughout the entire sequence.Sequence Cis a little different because we need to add -2 to the first number to get the second number.This too works for any pair of consecutive numbers.The fourth number plus -2 is the fifth number: 14 + (-2) = 12.Because these sequences behave according to this simple rule of addiing a constant number to one term to get to another, they are called arithmetic sequences.So that we can examine these sequences to greater depth, we must know that the fixed numbers that bind each sequence together are called thecommon differences. Mathematicians use the letterdwhen referring to these difference for this type of sequence.Mathematicians also refer to generic sequences using the letteraalong with subscripts that correspond to the term numbers as follows:This means that if we refer to the fifth term of a certain sequence, we will label it a 5.a 17is the 17th term.This notation is necessary for calculating nth terms, or a n, of sequences.Thed -value can be calculated by subtracting any two consecutive terms in an arithmetic sequence.where n is any positive integer greater than 1.Remember, the letterdis used because this number is called thecommon difference.When we subtract any two adjacent numbers, the right number minus the left number should be the same for any two pairs of numbers in an arithmetic sequence.
To determine any number within an arithmetic sequence, there are two formulas that can be utilized.Here is therecursive rule.The recursive rule means to find any number in the sequence, we must add the common difference to the previous number in this list.Let us say we were given this arithmetic sequence.
First, we would identify the common difference.We can see the common difference is 4 no matter which adjacent numbers we choose from the sequence.To find the next number after 19 we have to add 4.19 + 4 = 23.So, 23 is the 6th number in the sequence.23 + 4 = 27; so, 27 is the 7th number in the sequence, and so on.What if we have to find the 724th term?This method would force us to find all the 723 terms that come before it before we could find it.That would take too long.So, there is a better formula.It is called theexplicit rule.So, if we want to find the 724th term, we can use thisexplicit rule.Our n-value is 724 because that is the term number we want to find.The d-value is 4 because it is thecommon difference.Also, the first term, a 1, is 3.The rule gives us a 724= 3 + (724 – 1)(4) = 3 + (723)(4) = 3 + 2892 = 2895.
Each arithmetic sequence has its own unique formula.The formula can be used to find any term we with to find, which makes it a valuable formula.To find these formulas, we will use theexplicit rule.Let us also look at the following examples.Example 1 : Let’s examinesequence Aso that we can find a formula to express its nth term.If we match each term with it’s corresponding term number, we get:

Observations from experiments or measurements of natural occurrences are frequently used to generate numerical sequences. Such sequences can be made up of random numbers, although they are more typically made up of arithmetic or other ordered lists of numbers than they are of random numbers. The distinction between arithmetic and geometric sequences is that they share a common factor rather than a common difference, as in the case of geometric sequences. When a new term is introduced, instead of having a number added or removed for each term, a number is multiplied or divided for each term.

In contrast, some sequences are governed by entirely distinct principles.

1, 1, 2, 3, 5, 8, and so on are the numbers in this series.

Calculus sequences are straightforward, yet they have a variety of practical uses.

We can identify a few facts about it.Its first term, a 1, is 26.Itscommon differenceor d-value is 5.We can substitute this information into theexplicit rule.a n= a 1+ (n – 1)da n= 26 + (n – 1)(5)a n= 26 + 5n – 5a n= 5n + 21Now, we can use this formula to find its 14th term, like so. a n= 5n + 21a 14= 5(14) + 21a 14= 70 + 21a 14= 91ideo:Finding the nth Term of an Arithmetic Sequence uizmaster:Finding Formula for General Term
It may be necessary to calculate the number of terms in a certain arithmetic sequence. To do so, we would need to know two things.We would need to know a few terms so that we could calculate the common difference and ultimately the formula for the general term.We would also need to know the last number in the sequence.Once we know the formula for the general term of a sequence and the last term, the procedure involves the use of algebra.Use the two examples below to see how it is done.Example 1 : Find the number of terms in the sequence 5, 8, 11, 14, 17,., 47.This issequence A.In theprevious section, we found the formula to be a n= 3n + 2 for this sequence.We will use this along with the fact the last number, a n, is 47.We will plug this into the formula, like so.a n= 3n + 247 = 3n + 245 = 3n15 = nn = 15This means there are 15 numbers in this arithmetic sequence.Example 2 : Find the number of terms in the arithmetic sequence 20, 18, 16, 14, 12,.,-26.Our first task is to find the formula for this sequence given a 1= 20 and d = -2.We will substitute this information into theexplicit rule, like so.a n= a 1+ (n – 1)da n= 20 + (n – 1)(-2)a n= 20- 2n + 2a n= -2n + 22Now we can use this formula to find the number of terms in the sequence.Keep in mind, the last number in the sequence, a n, is -26.Substituting this into the formula gives us.a n= -2n + 22-26 = -2n + 22-48 = -2n24 = nn = 24This means there are 24 numbers in the arithmetic sequence.
Given our generic arithmeticsequence.we can add the terms, called aseries, as follows.There exists a formula that can add such a finite list of these numbers.It requires three pieces of information.The formula is.where S nis the sum of the first n numbers, a 1is the first number in the sequence and a nis the nth number in the sequence.If you would like to see a derivation of this arithmetic series sum formula, watch this video.ideo:Arithmetic Series: Deriving the Sum FormulaUsually problems present themselves in either of two ways.Either the first number and the last number of the sequence are known or the first number in the sequence and the number of terms are known.The following two problems will explain how to find a sum of a finite series.Example 1 : Find the sum of the series 5 + 8 + 11 + 14 + 17 +. + 128.In order to use the sum formula.We need to know a few things.We need to know n, the number of terms in the series.We need to know a 1, the first number, and a n, the last number in the series.We do not know what the n-value is.This is where we must start.To find the n-value, let’s use the formula for the series.We already determined the formula for the sequence in a previous section.We found it to be a n= 3n + 2.We will substitute in the last number of the series and solve for the n-value.a n= 3n + 2128 = 3n + 2126 = 3n42 = nn = 42There are 42 numbers in the series.We also know the d = 3, a 1= 5, and a 42= 128.We can substitute these number into the sum formula, like so.S n= (1/2)n(a 1+ a n)S 42= (1/2)(42)(5 + 128)S 42= (21)(133)S 42= 2793This means the sum of the first 42 terms of the series is equal to 2793.Example 2 : Find the sum of the first 205 multiples of 7.First we have to figure out what our series looks like.We need to write multiples of seven and add them together, like this.7 + 14 + 21 + 28 +. +?To find the last number in the series, which we need for the sum formula, we have to develop a formula for the series.So, we will use theexplicit ruleor a n= a 1+ (n – 1)d.We can also see that d = 7 and the first number, a 1, is 7.a n= a 1+ (n – 1)da n= 7 + (n – 1)(7)a n= 7 + 7n – 7a n= 7nNow we can find the last term in the series.We can do this because we were told there are 205 numbers in the series.We can find the 205th term by using the formula.a n= 7na n= 7(205)a n= 1435This means the last number in the series is 1435.It means the series looks like this.7 + 14 + 21 + 28 +. + 1435To find the sum, we will substitute information into the sum formula. We will substitute a 1= 7, a 205= 1435, and n = 205.S n= (1/2)n(a 1+ a n)S 42= (1/2)(205)(7 + 1435)S 42= (1/2)(205)(1442)S 42= (1/2)(1442)(205)S 42= (721)(205)S 42= 147805This means the sum of the first 205 multiples of 7 is equal to 147,805.

7.2 – Arithmetic Sequences

Experiments or measurements of natural events are frequently used to generate numerical sequences. Such sequences can be made up of random numbers, although they are more frequently made up of arithmetic or other ordered lists of integers. Geometric sequences, as opposed to arithmetic sequences, vary in that they share a common factor rather than a common difference. A number is multiplied or divided for each new phrase instead of being added or removed for each new word. As a geometric series with a common factor of 2, a sequence that is 10, 12, 14,.

Other sequences are governed by entirely other laws.

The numbers are in the following order: 1, 1, 2, 3, 5, 8.

Arithmetic sequences are straightforward, yet they have practical applications in everyday life.

Common Difference

The common difference is named as such since it is shared by all subsequent pairs of words and is thus referred to as such. It is indicated by the letter d. If the difference between consecutive words does not remain constant throughout time, the sequence is not mathematical in nature.

The common difference can be discovered by removing the terms from the sequence that are immediately preceding them. The following is the formula for the common difference of an arithmetic sequence: d = an n+1- a n

General Term

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

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

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

Partial Sum of an Arithmetic Sequence

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

  1. 2*S 5= (2+14) + (5+11) + (8+8) + (11+5) + (14+2) = (2+14) + (5+11) + (8+8) + (11+5) + (14+2) Take note that each of the amounts on the right-hand side is a multiple of 16.
  2. 2*S 5 = 5*(2 + 14) = 2*S 5 Finally, divide the total item by two to obtain the amount, not double the sum as previously stated.
  3. This would be 5/2 * (16) = 5(8) = 40 as a total.
  4. The number 5 refers to the fact that there were five terms, n.
  5. In this case, we added the total twice and it will always be a 2.
  6. Another formula for the n th partial sum of an arithmetic series is occasionally used in conjunction with the previous one.

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

Example

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

2+2=14+11+8+5+2=5+2=14+11+8+5+2=5 Now, combine the results of the two equations.

Instead of writing 16 (the sum of the first and final terms) five times, we may express it as 5 * 16 or 5 * (2 + 14), which is the same as 5 * (first and last terms) plus 5.

Once you’ve finished, divide the total by two to obtain the amount, not twice the sum.

To calculate this, multiply 5/2 by (16), which is 5(8).

Five words were used in all, therefore the number 5.

In this case, we added the total twice and it will always equal a 2.

Another formula for the n th partial sum of an arithmetic series is sometimes employed, and it is denoted by the symbol To produce it, substitute the formula for the general term into the previous formula, which has been simplified.

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

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