# What Is Arithmetic Sequences? (Best solution)

an+1= an + d, d — . : , d.

## How do you explain an arithmetic sequence?

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

## What is arithmetic sequence and example?

An arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term. For example in the arithmetic sequence 3, 9, 15, 21, 27, the common difference is 6. If we add or subtract by the same number each time to make the sequence, it is an arithmetic sequence.

## Why is it an arithmetic sequence?

What is an arithmetic sequence? For many of the examples above, the pattern involves adding or subtracting a number to each term to get the next term. Sequences with such patterns are called arithmetic sequences. In an arithmetic sequence, the difference between consecutive terms is always the same.

## What is the meaning of arithmetic means?

The arithmetic mean is the simplest and most widely used measure of a mean, or average. It simply involves taking the sum of a group of numbers, then dividing that sum by the count of the numbers used in the series. For example, take the numbers 34, 44, 56, and 78.

## 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.

## How are arithmetic sequences related to real life?

Arithmetic sequences are used in daily life for different purposes, such as determining the number of audience members an auditorium can hold, calculating projected earnings from working for a company and building wood piles with stacks of logs.

## 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

• 1, 4, 7, 10, 13, 16, 19, 22, and 25 are the first four digits of the number 1. The numbers in this sequence are separated by a factor of three. Each time the pattern is repeated, the last number is increased by three, as follows: The following is a general arithmetic sequence that we might write:

### 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:

• 3, 8, 13, 18, 23, 28, 33, and 38 are the numbers 3, 8, 13, 18, 23, 28, 33, and 38 respectively. Each number in this series differs by a factor of five. There are two values associated with aandd, which are:

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.

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

## 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)

Check out why the formula works since we’ll be employing an unusual “trick” that’s well worth your time to learn about. As a starting point, we’ll refer to the entire total as “S.” In S, the sum of the squares of the first and second roots of the first and second roots of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root of the second root Afterwards, rewrite S in the other direction: The formula for S is: (a + n1)d)+ (a + n2)d)+.

+(a + d)+.+(a + d)+. S = (a + n1)d)+. Now, term by term, combine the following two phrases.

## Arithmetic Sequence – Formula, Meaning, Examples

Check out why the formula works since we’ll be employing an unusual “trick” that’s well worth understanding. First, we shall 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 + (n2)d) + (a + (n1)d) + (a + (n1)d) + (a + (n2)d) + (a + (n1)d) + Afterwards, rewrite S in the opposite order: S = (a + (n1)d)+ (a + (n2)d)+ (a + (n3)d)+ (a + (n2)d)+. +(a + d)+a Now, term by term, combine the following two:

• The formula for determining the nth term of an arithmetic series. An arithmetic series has n terms, and the sum of the first n terms is determined by the following formula:

Let’s look at the definition of an arithmetic sequence, as well as arithmetic sequence formulae, derivations, and a slew of other examples to get us started.

 1 What is an Arithmetic Sequence? 2 Terms Related to Arithmetic Sequence 3 Nth Term of Arithmetic Sequence Formula 4 Sum of Arithmetic sequence Formula 5 Arithmetic Sequence Formulas 6 Difference Between Arithmetic and Geometric Sequence 7 FAQs on Arithmetic sequence

## What is an Arithmetic Sequence?

There are two ways in which anarithmetic sequence can be defined. When the differences between every two succeeding words are the same, it is said to be in sequence (or) Every term in an arithmetic series is generated by adding a specified integer (either positive or negative, or zero) to the term before it. Here is an example of an arithmetic sequence.

### Arithmetic Sequences Example

For example, consider the series 3, 6, 9, 12, 15, which is an arithmetic sequence since every term is created by adding a constant number (3) to the term immediately before that one. Here,

• A = 3 for the first term
• D = 6 – 3 for the common difference
• 12 – 9 for the second term
• 15 – 12 for the third term
• A = 3 for the third term

As a result, arithmetic sequences can be expressed as a, a + d, a + 2d, a + 3d, and so forth. Let’s use the previous scenario as an example of how to test this pattern. a, a + d, a + 2d, a + 3d, a + 4d,. = 3, 3 + 3, 3 + 2(3), 3 + 3(3), 3 + 4(3),. = 3, 6, 9, 12, 15,. = 3, 6, 9, 12, 15,. = 3, 6, 9, 12, 15,. = 3, 6, 9, 12, 15,. Here are a few more instances of arithmetic sequences to consider:

• 5, 8, 11, 14,
• 80, 75, 70, 65, 60,
• 2/2, 3/2, 2/2,
• -2, -22, -32, -42,
• 5/8, 11/14,

The terms of an arithmetic sequence are often symbolized by the letters a1, a2, a3, and so on. Arithmetic sequences are discussed in the following way, according to the vocabulary we employ.

### First Term of Arithmetic Sequence

The first term of an arithmetic sequence is, as the name implies, the first integer in the sequence. It is often symbolized by the letters a1 (or) a. For example, the first word in the sequence 5, 8, 11, 14, is the number 5. Specifically, a1 = 6 (or) a = 6.

### Common Difference of Arithmetic Sequence

The addition of a fixed number to each preceding term in an arithmetic series, with one exception (the first term), has previously been demonstrated in prior sections. The “fixed number” in this case is referred to as the “common difference,” and it is symbolized by the letter d. The formula for the common difference isd = a – an1.

## Nth Term of Arithmetic Sequence Formula

In such case, the thterm of an arithmetic series of the form A1, A2, A3,. is given byan = a1 + (n-1) d. This is also referred to as the broad word for the arithmetic sequence in some circles. This comes immediately from the notion that the arithmetic sequence a1, a2, a3,. = a1, a1 + d, a1 + 2d, a1 + 3d,. = a1, a1 + d, a1 + 3d,. = a1, a1 + 3d,. = a1, a1 + 3d,. Several arithmetic sequences are shown in the following table, along with the first term, the common difference, and the subsequent n thterms.

Arithmetic sequence First Term(a) Common Difference(d) n thtermaₙ = a₁ + (n – 1) d
80, 75, 70, 65, 60,. 80 -5 80 + (n – 1) (-5)= -5n + 85
π/2, π, 3π/2, 2π,. π/2 π/2 π/2 + (n – 1) (π/2)= nπ/2
-√2, -2√2, -3√2, -4√2,. -√2 -√2 -√2 + (n – 1) (-√2)= -√2 n

### Arithmetic Sequence Recursive Formula

It is possible to utilize the following formula for finding the nthterm of an arithmetic series in order to discover any term of that sequence if the values of ‘a1′ and’d’ are known, however this is not recommended. One further method of determining what term is the n thterm is to utilize the ” recursive formula of an arithmetic sequence “. This formula may be used to determine the next term (an) of an arithmetic sequence given both its preceding term (an1) and the value of the variable ‘d’ are known.

Example: If a19 = -72 and d = 7, find the value of a21 in an arithmetic sequence. Solution: a20 = a19 + d = -72 + 7 = -65 is obtained by applying the recursive formula. a21 = a20 + d = -65 + 7 = -58; a21 = a20 + d = -65 + 7 = -58; a21 = a20 + d = -65 + 7 = -58; As a result, the value of a21 is -58.

## Sum of Arithmetic sequence Formula

To obtain the sum of the first n terms of an arithmetic sequence, the sum of the arithmetic sequence formula is employed. Consider the following arithmetic sequence: a1 (or ‘a’) is the first term, and d is the common difference between the first and second terms. Sn is the symbol for the sum of the first n terms in the expression. Then

• To obtain the sum of the first n terms of an arithmetic sequence, the sum of the arithmetic sequence formula is utilized. Consider the following arithmetic sequence: a1 (or ‘a’) is the first term, and d is the common difference between the first and second terms: Sn is the symbol for the sum of its first n terms. Then
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Example Ms. Natalie makes \$200,000 each year, with an annual pay rise of \$25,000 in addition to that. So, how much money does she have at the conclusion of the first five years of her career? Solution In Ms. Natalie’s first year of employment, she earns a sum equal to a = 2,000,000. The annual increase is denoted by the symbol d = 25,000. We need to figure out how much money she will make in the first five years. As a result, n = 5. In the sum sum of arithmetic sequence formula, substituting these numbers results in Sn = n/2 Sn = 5/2(2(200000) + (5 – 1)(25000), which is 5/2 (400000 +100000), which is equal to 5/2 (500000), which is equal to 1250000.

We may modify this formula to be more useful for greater values of the constant ‘n.’

### Sum of Arithmetic Sequence Proof

Consider the following arithmetic sequence: a1 is the first term, and d is the common difference between the two terms. The sum of the first ‘n’ terms of the series is given bySn = a1 + (a1 + d) + (a1 + 2d) +. + an, where Sn = a1 + (a1 + d) + (a1 + 2d) +. + an. (1) Let us write the same total from right to left in the same manner (i.e., from the n thterm to the first term). (an – d) + (an – 2d) +. + a1. Sn = a plus (an – d) plus (an – 2d) +. + a1. (2)By combining (1) and (2), all words beginning with the letter ‘d’ are eliminated.

+ (a1 + an) 2Sn = n (a1 + an) = n (a1 + an) Sn =/2 is a mathematical expression.

## Arithmetic Sequence Formulas

The following are the formulae that are connected to the arithmetic sequence.

• There is a common distinction, the n-th phrase, a = (a + 1)d
• The sum of n terms, Sn =/2 (or) n/2 (2a + 1)d
• The n-th term, a = (a + 1)d
• The n-th term, a = a + (n-1)d

## Difference Between Arithmetic and Geometric Sequence

The following are the distinctions between arithmetic sequence and geometric sequence:

 Arithmetic sequences Geometric sequences In this, the differences between every two consecutive numbers are the same. In this, theratiosof every two consecutive numbers are the same. It is identified by the first term (a) and the common difference (d). It is identified by the first term (a) and the common ratio (r). There is a linear relationship between the terms. There is an exponential relationship between the terms.

Notes on the Arithmetic Sequence that are very important

• The Arithmetic Sequence: Important Points to Keep in Mind

Arithmetic Sequence-Related Discussion Topics

• Sequence Calculator, Series Calculator, Arithmetic Sequence Calculator, Geometric Sequence Calculator are all terms used to refer to the same thing.

## Solved Examples on Arithmetic Sequence

1. Examples: Find the nth term in the arithmetic sequence -5, -7/2, -2 and the nth term in the arithmetic sequence Solution: The numbers in the above sequence are -5, -7/2, -2, and. There are two terms in this equation: the first is equal to -5, and the common difference is equal to -(7/2) – (-5) = -2 – (-7/2) = 3/2. The n thterm of an arithmetic sequence can be calculated using the formulaan = a + b. (n – 1) dan = -5 +(n – 1) (3/2)= -5+ (3/2)n – 3/2= 3n/2 – 13/2 = dan = -5 +(n – 1) (3/2)= dan = -5 +(n – 1) (3/2)= dan = -5 +(n – 1) (3/2)= dan = -5 +(3/2)n – 3/2= dan = -5 +(3/2)n – 3/2= dan = Example 2:Which term of the arithmetic sequence -3, -8, -13, -18, the answer is: the specified arithmetic sequence is: 3, 8, 13, 18, and so on. The first term is represented by the symbol a = -3. The common difference is d = -8 – (-3) = -13 – (-8) = -5. The common difference is d = -8 – (-3) = -13 – (-8) = -5. It has been established that the n thterm is a = -248. All of these values should be substituted in the n th l term of an arithmetic sequence formula,an = a + b. (n – 1) d-248 equals -3 plus (-5) (n – 1) the sum of -248 and 248 equals 3 -5n, and the sum of 5n and 250 equals -5nn equals 50. Answer: The number 248 represents the 50th phrase in the provided sequence.

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## FAQs on Arithmetic sequence

An arithmetic sequence is a sequence of integers in which every term (with the exception of the first term) is generated by adding a constant number to the preceding term. For example, the arithmetic sequence 1, 3, 5, 7, is an arithmetic sequence because each term is created by adding 2 (a constant integer) to the term before it.

### What are Arithmetic Sequence Formulas?

Here are the formulae connected to an arithmetic series where a1 (or a) is the first term and d is the common difference: a1 (or a) is the first term, and d is the common difference:

• When we look at the common difference, it is second term minus first term. The n thterm of the series is defined as a = a + (n – 1)d
• Sn =/2 (or) n/2 (2a + (n – 1)d) is the sum of the n terms in the sequence.

### How to Find An Arithmetic Sequence?

Whenever the difference between every two successive terms of a series is the same, then the sequence is said to be an arithmetic sequence. For example, the numbers 3, 8, 13, and 18 are arithmetic because

### What is the n thterm of an Arithmetic Sequence Formula?

The n thterm of arithmetic sequences is represented by the expression a = a + (n – 1) d. The letter ‘a’ stands for the first term, while the letter ‘d’ stands for the common difference.

### What is the Sum of an Arithmetic Sequence Formula?

Arithmetic sequences with a common difference ‘d’ and the first term ‘a’ are denoted by Sn, and we have two formulae to compute the sum of the first n terms with the common difference ‘d’.

### What is the Formula to Find the Common Difference of Arithmetic sequence?

As the name implies, the common difference of an arithmetic sequence is the difference between every two of its consecutive (or consecutively occurring) terms. Finding the common difference of an arithmetic series may be calculated using the formula: d = a – an1.

### How to Find n in Arithmetic sequence?

When we are asked to find the number of terms (n) in arithmetic sequences, it is possible that part of the information about a, d, an, or Sn has already been provided in the problem. We will simply substitute the supplied values in the formulae of an or Sn and solve for n as a result of this.

### How To Find the First Term in Arithmetic sequence?

The number that appears in the first position from the left of an arithmetic sequence is referred to as the first term of the sequence. It is symbolized by the letter ‘a’. If the letter ‘a’ is not provided in the problem, then the problem may contain some information concerning the letter d (or) the letter a (or) the letter Sn. We shall simply insert the specified values in the formulae of an or Sn and solve for a by dividing by two.

### What is the Difference Between Arithmetic Sequence and Arithmetic Series?

When it comes to numbers, an arithmetic sequence is a collection in which all of the differences between every two successive integers are equal to one, and an arithmetic series is the sum of a few or more terms of an arithmetic sequence.

### What are the Types of Sequences?

In mathematics, there are three basic types of sequences. They are as follows:

• The arithmetic series, the geometric sequence, and the harmonic sequence are all examples of sequences.

### What are the Applications of Arithmetic Sequence?

Here are some examples of applications: The pay of a person who receives an annual raise of a fixed amount, the rent of a taxi that charges by the mile traveled, the number of fish in a pond that increases by a certain number each month, and so on are examples of steady increases.

### How to Find the n thTerm in Arithmetic Sequence?

The following are the actions to take in order to get the n thterm of arithmetic sequences:

• Identify the first term, a
• The common difference, d
• And the last term, e. Choose the word that you wish to use. n, to be precise. All of them should be substituted into the formula a = a + (n – 1) d

### How to Find the Sum of n Terms of Arithmetic Sequence?

To get the sum of the first n terms of arithmetic sequences, use the following formula:

• Identify the initial term (a)
• The common difference (d)
• And the last term (e). Determine which phrase you wish to use (n)
• All of them should be substituted into the formula Sn= n/2(2a + (n – 1)d)

## Arithmetic progression – Wikipedia

The evolution of mathematical operations The phrase “orarithmetic sequence” refers to a sequence of integers in which the difference between successive words remains constant. Consider the following example: the sequence 5, 7, 9, 11, 13, 15,. is an arithmetic progression with a common difference of two. As an example, if the first term of an arithmetic progression is and the common difference between succeeding members is, then in general the -th term of the series () is given by:, and in particular, A finite component of an arithmetic progression is referred to as a finite arithmetic progression, and it is also referred to as an arithmetic progression in some cases.

Anarithmetic series is the sum of a finite arithmetic progression, and it is composed of the numbers 0 through 9.

## 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.

However, it is possible that the intersection of infinitely many infinite arithmetic progressions is a single number rather than being an endless progression in and of itself.

## 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.

• 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.
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