What Is An Arithmetic Equation? (Solution found)

An arithmetic formula is a sequence of numbers that is ordered with a specific pattern. Each successive number is the sum of the previous number and a constant. The constant is the same for every term in the sequence and is called the common difference.

Contents

What type of equation is arithmetic?

An arithmetic sequence is a sequence in which the difference between each consecutive term is constant. An arithmetic sequence can be defined by an explicit formula in which an = d (n – 1) + c, where d is the common difference between consecutive terms, and c = a1.

What is simple arithmetic equations?

The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division. The basic arithmetic properties are the commutative, associative, and distributive properties.

WHAT IS A in an 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 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 is the general formula for quadratic equation?

The quadratic formula is a general formula used for solving the quadratic equation: x = − b ± b 2 − 4 a c 2 a. Note that: if x2 = k, where k ≥ 0, then or same as x = ± k, where ± means “plus or minus.”

What is an equation example?

An equation is a mathematical sentence that has two equal sides separated by an equal sign. 4 + 6 = 10 is an example of an equation. We can see on the left side of the equal sign, 4 + 6, and on the right hand side of the equal sign, 10. For example, 12 is the coefficient in the equation 12n = 24.

How many math equations are there?

There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by an equals sign (“=”).

What are the 4 types of equations?

Different Types of Equations

  • Linear Equation.
  • Radical Equation.
  • Exponential Equation.
  • Rational Equation.

How do you find a simple equation?

Addition and subtraction equations

  1. Example 1. Solve for x. x + 8 = 12. To solve the equation x + 8 = 12, you must get x by itself on one side.
  2. Example 2. Solve for y. y – 9 = 25. To solve this equation, you must get y by itself on one side.
  3. Example 3. Solve for x. x + 15 = 6. To solve, subtract 15 from both sides.

What is the formula in finding the sum of an arithmetic sequence?

The sum of an arithmetic sequence is given by Sn=n∑i=1ai=n2(a1+an).

Arithmetic Sequence Formula – What is Arithmetic Sequence Formula? Examples

Calculating the nth term of an arithmetic progression is accomplished through the use of the arithmetic sequence formula. The arithmetic sequence is a series in which the common difference between any two succeeding terms remains constant throughout the sequence. In order to discover any term in the arithmetic sequence, we may use the arithmetic sequence formula, which is defined as follows: Let’s look at several solved cases to better grasp the arithmetic sequence formula.

What Is the Arithmetic Sequence Formula?

An Arithmetic sequence has the following structure: a, a+d, a+2d, a+3d, and so on up to n terms. In this equation, the first term is called a, the common difference is called d, and n = the number of terms is written as n. Recognize the arithmetic sequence formulae and determine the AP, first term, number of terms, and common difference before proceeding with the computation. Various formulae linked with an arithmetic series are used to compute the n thterm, total, or common difference of a given arithmetic sequence, depending on the series in question.

Arithmetic Sequence Formula

The arithmetic sequence formula is denoted by the notation Formula 1 is a racing series that takes place on the track. The arithmetic sequence formula is written as (a_ =a_ +(n-1) d), where an is the number of elements in the series.

  • A_ is the n th term
  • A_ is the initial term
  • And d is the common difference.

The n thterm formula of anarithmetic sequence is sometimes known as the n thterm formula of anarithmetic sequence. For the sum of the first n terms in an arithmetic series, the formula is (S_ = frac), where S is the number of terms.

  • (S_ ) is the sum of n terms
  • (S_ ) is the sum of n terms
  • A is the initial term, and d is the difference between the following words that is common to all of them.

Formula 3: The formula for determining the common difference of an AP is given as (d=a_ -a_ )where, a_ is the AP’s initial value and a_ is the common difference of the AP.

  • There are three terms in this equation: nth term, second last term, and common difference between the consecutive terms, denoted by the letter d.

Formula 4: When the first and last terms of an arithmetic progression are known, the sum of the first n terms of the progression is given as, (s_ = fracleft )where, and

  • (S_ ) is the sum of the first n terms
  • (a_ ) is the last term
  • And (a_ ) is the first term.

Applications of Arithmetic Sequence Formula

Each and every day, and sometimes even every minute, we employ the arithmetic sequence formula without even recognizing it. The following are some examples of real-world uses of the arithmetic sequence formula.

  • Arranging the cups, seats, bowls, or a house of cards in a towering fashion
  • There are seats in a stadium or a theatre that are set up in Arithmetic order
  • The seconds hand on the clock moves in Arithmetic Sequence, as do the minutes hand and the hour hand
  • The minutes hand and the hour hand also move in Arithmetic Sequence. The weeks in a month follow the AP, and the years follow the AP as well. It is possible to calculate the number of leap years simply adding 4 to the preceding leap year. Every year, the number of candles blown on a birthday grows in accordance with the mathematical sequence

Consider the following instances that have been solved to have a better understanding of the arithmetic sequence formula. Do you want to obtain complicated math solutions in a matter of seconds? To get answers to difficult queries, you may use our free online calculator. Find solutions in a few quick and straightforward steps using Cuemath. Schedule a No-Obligation Trial Class.

Examples Using Arithmetic Sequence Formula

In the first example, using the arithmetic sequence formula, identify the thirteenth term in the series 1, 5, 9, and 13. Solution: To locate the thirteenth phrase in the provided sequence. Due to the fact that the difference between consecutive terms is the same, the above sequence is an arithmetic series. a = 1, d = 4, etc. Making use of the arithmetic sequence formula (a_ =a_ +(n-1) d) = (a_ =a_ +(n-1) d) = (a_ =a_ +(n-1) d) For the thirteenth term, n = 13(a_ ) = 1 + (13 – 1) 4(a_ ) = 1 + 4(a_ ) (12) The sum of 4(a_ ) and 48(a_ ) equals 49.

Example 2: Determine the first term in the arithmetic sequence in which the 35th term is 687 and the common difference between the two terms.

Solution: In order to locate: The first term in the arithmetic sequence is called the initial term.

Example 3: Calculate the total of the first 25 terms in the following sequence: 3, 7, 11, and so on.

In this case, (a_ ) = 3, d = 4, n = 25. The arithmetic sequence that has been provided is 3, 7, 11,. With the help of the Sum of Arithmetic Sequence Formula (S_ =frac), we can calculate the sum of the first 25 terms (S_ =frac) as follows: (25/2) = 25/2 102= 1275.

FAQs on Arithmetic Sequence Formula

It is referred to as arithmetic sequence formula when it is used to compute the general term of an arithmetic sequence as well as the sum of all n terms inside an arithmetic sequence.

What Is n in Arithmetic Sequence Formula?

It is important to note that in the arithmetic sequence formula used to obtain the generalterm (a_ =a_ +(n-1) d), n refers to how many terms are in the provided arithmetic sequence.

What Is the Arithmetic Sequence Formula for the Sum of n Terms?

The sum of the first n terms in an arithmetic series is denoted by the expression (S_ =frac), where (S_ ) =Sum of n terms, (a_ ) = first term, and (d) = difference between the first and second terms.

How To Use the Arithmetic Sequence Formula?

Determine whether or not the sequence is an AP, and then perform the simple procedures outlined below, which vary based on the values known or provided:

  • This is the formula for thearithmetic sequence: (a_ =a_ +(n-1) d), where a_ is a general term, a_ is a first term, and d is the common difference between the two terms. This is done in order to locate the general word inside the sequence. The sum of the first n terms in an arithmetic series is denoted by the symbol (S_ =frac), where (S_ ) =Sum of n terms, (a_ )=first term, and (d) represents the common difference between the terms. When computing the common difference of an arithmetic series, the formula is stated as, (d=a_ -a_ ), where a_ is the nth term, a_ is the second last term, and d is the common difference. Arithmetic progression is defined as follows: (s_ =fracleft) = Sum of first n terms, nth term, and nth term
  • (s_ =fracright) = First term
  • (s_ =fracleft)= Sum of first two terms
  • And (s_ =fracright) = Sum of first n terms.

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.

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.

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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

In any mathematical progression, the standard deviation may be determined aswhere is the number of terms in the progression and is the average difference between terms. The standard deviation of an adiscrete uniform distribution is extremely similar to the expression.

History

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

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.”

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:

Using the Formula for Arithmetic Series

In the same way that we looked at different sorts of sequences, we will look at different forms of series. Remember that anarithmetic sequence is a series in which the difference between any two successive terms is equal to the common difference,d, and thus The sum of the terms of an arithmetic sequence is referred to as an anarithmetic series in mathematical jargon. It is possible to represent the sum of the firstnterms of an arithmetic series as: = +left( +dright)+left( +2dright)+.+left( -dright)+ +left( -dright).

The sum of the firstnterms of an arithmetic series may be calculated by adding these two expressions for the sum of the firstnterms of an arithmetic series together.

Fractal = +left( +dright)+left( +2dright)+.+left( -dright)+ hfill + = +left( -dright)+left( -2dright)+.+left( +dright)+ hfill = +left( +dright)+left( -2dright)+.

To determine the formula for the sum of the firstnterms of an arithmetic series, we divide the number by two.

A General Note: Formula for the Sum of the FirstnTerms of an Arithmetic Series

The sum of the terms of an arithmetic sequence is known as an anarithmetic series. It is written as =frac + right) for the sum of the firstnterms of an arithmetic sequence:

How To: Given terms of an arithmetic series, find the sum of the firstnterms.

  1. Identify and
  2. Determinen
  3. Substitute values for text , andninto the formula =frac + right)n
  4. Substitute values for text , andninto the formula =frac + right)n Make it easier to find_

Example 2: Finding the FirstnTerms of an Arithmetic Series

Calculate the sum of each arithmetic series in the given time frame.

Solution

  1. We are given the numbers_ =5 and_ =32. To findn=10, count the number of phrases in the sequence to get at n=10. Simplify the formula by substituting values for, text, andninto the equation. begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin Make use of the formula for the general term of an arithmetic series to arrive at the answer. begin = +left begin dhfill -50=20+left(n – 1right)+left(n – 1right)+left(n – 1right)+left(n – 1right)+left(n – 1right)+left(n – 1right)+left(n – 1right)+left (-5right) left(n + 1 right)hfill -70=left(n – 1 right)hfill -70=hfill -70=hfill -70=hfill -70=hfill -70=hfill -70 (-5right) hfill 14=n – 1hfill 15=nhfill nhfill nhfill nhfill end Substitute values for_, _text ninto the formula to make it easier to understand. begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin begin Begin_ =3k – 8hfill text_ =3k – 3hright Fill in the blanks (1right) -8=-5hfill hfill hend We are given the information thatn=12. To find_, enter k=12 into the explicit formula that has been provided. fill in the blanks with text_ =3k – 8hfill in the blanks with text_ =3left(12right)-8=28hfill in the blanks with end text_ =3k – 8hfill in the blanks with end text_ Simplify the formula by substituting values for the variables_, _, andn. hfill =frac + right)hfill =frac=138hfill end
  2. Hfill =frac=138hfill end

To get the sum of each arithmetic series, use the formula provided.

Try It 2

solution text text text text text text text text text text text text text text text text text text text text text text text text word text text

Try It 3

Dots in text text Dots in text Solution

Try It 4

sum 5 – 6k Solution to the problem

Example 3: Solving Application Problems with Arithmetic Series

A lady is able to walk a half-mile on Sunday after having minor surgery, which she does on Saturday. Every Sunday, she adds an additional quarter-mile to her daily stroll. What do you think the total number of kilometers she has walked will be after 8 weeks?

Solution

This problem may be represented by an arithmetic series with_ =fracandd=frac as the first and second terms. The total number of kilometers walked after 8 weeks is what we are seeking for; thus, we know thatn=8 and that we are looking for We may use the explicit formula for an arithmetic series to get the value of . commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commencement commence We can now apply the arithmetic series formula to our advantage.

hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill hfill She will have walked a total of 11 miles by the time she is through.

Try It 5

In the first week of June, a man receives $100 in pay. The amount he makes each week is $12.50 greater than the previous week. How much money has he made after 12 weeks of work? Solution

Formulas for Arithmetic Sequences

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

Using Explicit Formulas for Arithmetic Sequences

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

  1. Considering that the average difference is 50, the series represents a linear function with an associated slope of 50.
  2. You may also get the they-intercept by graphing the function and calculating the point at which a line connecting the points would intersect the vertical axis, as shown in the example.
  3. When working with sequences, we substitute _instead of y and ninstead of n.
  4. Using 50 as the slope and 250 as the vertical intercept, we arrive at this equation: = -50n plus 250 To create an explicit formula for an arithmetic series, we do not need to identify the vertical intercept of the sequence.

For this sequence, there is another explicit formula, which is_ =200 – 50left(n – 1right), which may be simplified to_ =-50n+250.

A General Note: Explicit Formula for an Arithmetic Sequence

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

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

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

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

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

Try It

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

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

A General Note: Recursive Formula for an Arithmetic Sequence

In the case of an arithmetic sequence with common differenced, the recursive formula is as follows: the beginning of the sentence = +dnge 2 the finish of the sentence

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

  1. In the case of an arithmetic series with a common differenced, the recursive formula is as follows: ( Beginning with the letter _, and ending with the letter _, we have the expression

Example: Writing a Recursive Formula for an Arithmetic Sequence

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

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

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

Try It

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

Find the Number of Terms in an Arithmetic Sequence

When determining the number of terms in a finite arithmetic sequence, explicit formulas can be employed to make the determination. Finding the common difference and determining the number of times the common difference must be added to the first term in order to produce the last term of the sequence are both necessary steps.

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

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

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

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

Try It

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

Solving Application Problems with Arithmetic Sequences

In many application difficulties, it is frequently preferable to begin with the term instead of_ as an introductory phrase. When solving these problems, we make a little modification to the explicit formula to account for the change in beginning terms. The following is the formula that we use: = +dn = = +dn

Example: Solving Application Problems with Arithmetic Sequences

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

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

Try It

A lady chooses to go for a 10-minute run every day this week, with the goal of increasing the length of her daily exercise by 4 minutes each week after that.

Create a formula to predict the timing of her run after n weeks has passed. In eight weeks, how long will her daily run last on average?

Contribute!

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

Arithmetic Series

Algebra Applied Mathematics is a branch of mathematics that is used in a variety of applications. Calculus and Statistical Analysis Discrete Mathematics is a branch of mathematics that deals with discrete events. The Mathematics of the Foundations Geometry The Background and Terminology Number Theory is a branch of mathematics that studies the relationships between numbers. Probability and Statistics are two terms that are used interchangeably. Recreational Mathematics is a term that refers to the study of mathematics for fun.

  • When you add up the terms of a sequence, you get an arithmetic series.
  • As a result, in the case of1″>, In order to find the sum of the series of the firstterms, we use thesumidentity function.
  • This is the approach Gauss employed as a schoolboy to solve the issue of summing the integers from 1 to 100, which had been assigned to him as “busy-work” by his mathematician instructor.
  • He has a few things on his mind (Burton 1989, pp.
  • 207).

See also

Arithmetic Progression, Common Difference, Geometric Series, Harmonic Series, and Prime Numbers are all examples of functions. Iterative Progression in Arithmetic In the MathWorld classroom, you may learn more about this topic.

Explore with Wolfram|Alpha

M. Abramowitz and I. A. Stegun have published a paper in which they discuss their work (Eds.). This is the ninth printing of the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York, p. 10, 1972. W. H. Beyer & Sons, Inc. (Ed.). 1987, page 8 of CRC Standard Mathematical Tables, 28th ed. Boca Raton, Florida: CRC Press. Allyn and Bacon published the fourth edition of Burton’s Elementary Number Theory in Boston, Massachusetts in 1989. “The Arithmetical Progression,” written by R.

  • Robbins.
  • 1.2.2 in What Is Mathematics?
  • “The Man Who Loved Only Numbers”: The Story of Paul Erdos and the Search for Mathematical Truth, by P.
  • Hyperion Publishing Company, New York, 1998.
  • Pappas’s book, The Joy of Mathematics, is available online.
  • 164, 1989.

Referenced on Wolfram|Alpha

Arithmetic Sequences are a type of series that may be used to solve problems.

Cite this as:

Arithmetic Sequences are a type of series that may be used to calculate numbers.

Subject classifications

It is the sum of the terms of an arithmetic sequence that is known as an arithmetic series. A geometric series is made up of the terms of a geometric sequence and is represented by the symbol. You can work with other sorts of series as well, but you won’t have much experience with them until you get to calculus. For the time being, you’ll most likely be collaborating with these two. How to deal with arithmetic series is explained and shown on this page, among other things. You can only take the “partial” sum of an arithmetic series for a variety of reasons that will be explored in greater detail later in calculus.

The following is the formula for the firstnterms of the anarithmeticsequence, starting with i= 1, and it is written: Content Continues Below The “2” on the right-hand side of the “equals” sign may be converted to a one-half multiplied on the parenthesis, which reveals that the formula for the total is, in effect,n times the “average” of the first and final terms, as seen in the example below.

The summation formula may be demonstrated via induction, by the way.

Find the35 th partial sum,S 35, of the arithmetic sequence with terms

The first thirty-five terms of this sequence are added together to provide the 35th partial sum of the series. The first few words in the sequence are as follows: Due to the fact that all of the words share a common difference, this is in fact an arithmetic sequence. The final term in the partial sum will be as follows: Plugging this into the formula, the 35 th partial sum is:Then my answer is:35 th partial sum:Then my answer is:35 th partial sum: S 35 = 350 S 35 = 350 If I had merely looked at the formula for the terms in the series above, I would have seen the common difference in the above sequence.

If we had used a continuous variable, such as the “x” we used when graphing straight lines, rather than a discrete variable, then ” ” would have been a straight line that rose by one-half at each step, rather than the discrete variable.

Find the value of the following summation:

It appears that each term will be two units more in size than the preceding term based on the formula ” 2 n– 5 ” for the then-thirteenth term. (Whether I wasn’t sure about something, I could always plug in some values to see if they were correct.) As a result, this is a purely arithmetic sum. However, this summation begins at n= 15, not at n= 1, and the summation formula is only applicable to sums that begin at n=1. So, how am I supposed to proceed with this summation? By employing a simple trick: The simplest approach to get the value of this sum is to first calculate the 14th and 47th partial sums, and then subtract the 14th from the 47th partial sum.

By doing this subtraction, I will have subtracted the first through fourteenth terms from the first through forty-seventh terms, and I will be left with the total of the fifteenth through forty-seventh terms, as shown in the following table.

These are the fourteenth and forty-seventh words, respectively, that are required: a14= 2(14) – 5 = 23a47= 2(47) – 5 = 89a14= 2(14) – 5 = 23a47= 2(47) – 5 = 89 With these numbers, I now have everything I need to get the two partial sums for my subtraction, which are as follows: I got the following result after subtracting: Then here’s what I’d say: As a side note, this subtraction may also be written as ” S 47 – S 14 “.

Don’t be shocked if you come into an exercise that use this notation and requires you to decipher its meaning before you can proceed with your calculations; this is common.

If you’re working with anything more complicated, though, it may be important to group symbols together in order to make the meaning more obvious. In order to do so correctly, the author of the previous exercise should have structured the summation using grouping symbols in the manner shown below:

Find the value ofnfor which the following equation is true:

Knowing that the first term has the value a1= 0.25(1) + 2 = 2.25, I may proceed to the second term. It appears from the formula that each term will be 0.25 units larger than the preceding term, indicating that this is an arithmetical series withd= 0.25, as shown in the diagram. The summation formula for arithmetical series therefore provides me with the following results: The number n is equal to 2.25 + 0.25 + 2 = 42n is equal to 0.25 + 4.25 + 42 = 420.25 n2+ 4.25 n– 42 = 0n2+ 17 n– 168 = 0(n+ 24)(n– 7 = 0n2+ 17 n– 168 = 0(n+ 24)(n– 7).

Then n= 7 is the answer.

However, your instructor may easily assign you a summation that needs you to use, say, eighty-six words or a thousand terms in order to arrive at the correct total.

As a result, be certain that you are able to do the calculations from the formula.

Find the sum of1 + 5 + 9 +. + 49 + 53

After looking through the phrases, I can see that this is, in fact, an arithmetic sequence: The sum of 5 and 1 equals 49 and 5 equals 453 and 49 equals 4. The reason for this is that they won’t always inform me, especially on the exam, what sort of series they’ve given me. (And I want to get into the habit of checking this way.) They’ve provided me the first and last terms of this series, however I’m curious as to how many overall terms there are in this series. This is something I’ll have to sort out for myself.

After plugging these numbers into the algorithm, I can calculate how many terms there are in total: a n=a1+ (n–1) d 53 = 1 + (n–1) a n=a1+ (n–1) (4) 53 = 1 + 4 n– 453 = 4 n– 356 = 4 n– 14 =n 53 = 1 + 4 n– 453 = 4 n– 356 = 4 n– 14 =n There are a total of 14 words in this series.

+ 49 + 53 = 1 + 5 + 9 Then I’ll give you my answer: partial sum S 14 = 378 S 14= 378 After that, we’ll look at geometric series.

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.

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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

The succession of numbers in anarithmetic A series of numbers in which each succeeding number is the sum of the preceding number plus certain constants, for example, the development of numbers in arithmetic terms 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., 2). an=an−1+d Number Sequences in Arithmetic As a result, the constant is referred to as the common difference since anan1=d.

An arithmetic sequence is, for example, the series of positive odd integers: 1, 3, 5, 7, 9,.

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. It is possible that the initial term of an arithmetic series will not be provided in some instances.

Example 4

All words that fall between a1=8 and a7=10 are to be discovered. as part of an arithmetic progression Or, to put it another way, locate all of the arithmetic means between the 1st and 7th term. Solution: First, identify the differences that are shared by all participants. Here, we are given the first and seventh terms: first and seventh terms an=a1+(n−1) d You may use the number n = 7 and the formula a1+(71)da7=a1+6d is used: a1+ (6d) Substitutea1=−8anda7=10 into the preceding equation, and then solve for the common differenced value 10=−8+6d18=6d3=d Use the first terma1=8 as a starting point for your next step.

An example is 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} According to mathematics, the number 7 equals 3 (7 11) and the number 10 equals 10.

It is possible that the initial term of an arithmetic series will not be provided in some circumstances.

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

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. Sn=n(a1+an)2S50=50.(a1+a50)2=50(4+249) 2=25(253)=6,325 Answer_S50=6,325

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.

where a1=26 and d=2.

As a result, the number of seats in each row may be calculated using the formulaan=2n+24.

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.

Calculate the sum of the first 60 terms of the following sequence of numbers: 5, 0, 5, 10, 15,.

Answer:S60=−8,550

Key Takeaways

  • If you’re looking for seats in an outdoor amphitheater, the first row has 26 seats, the second row has 28, the third row has 30, the fourth row has 32, and so on. What is the total number of seats available in the theater if there are 18 rows? Figure 9.2: The Roman Forum (Wikipedia) Solution: To begin, look for a formula that may be used to determine the number of seats in each given row of seating. Each row has a different number of seats, resulting in an order of: 26,28,30,… It is important to note that the difference between any two consecutive phrases is always two. An arithmetic progression is used to describe the series. An=(n1)d=26+(n1)d2=26+(n1)2=26+(n1)2=2n+24, wherea1=26andd=2. wherea1=26andd =2. As a result, an=2n+24 is used to calculate the number of seats in each row. This partial sum must be computed in order to get the overall seating capacity for all 18 rows. We require the following 18 thterms in order to do this: a1=26a18=2(18)+24=60 Calculate the 18thparticular sum as follows with this information: Sn=n(a1+an)2S18=18⋅(a1+a18)2=18(26+60) 2=9(86)=774 There are a total of 774 available seats. You should try it. The sum of the first 60 terms of the following sequence is to be calculated as follows: Five, zero, five, ten, fifteen. Answer_S60=−8,550

Topic Exercises

  1. Given the first term and common difference of an arithmetic series, write the first five terms of the sequence. Find a formula that describes the generic term. The values of a1 are 5 and 3
  2. 12 and 2
  3. 15 and 5
  4. 7 and 4 respectively
  5. 12 and 1
  6. A1=23 and 13 respectively
  7. 1 and 12 respectively
  8. A1=54 and 14. The values of a1 are 1.8 and 0.6
  9. 4.3 and 2.1
  10. And a1=5.4 and 2.1 respectively.
  1. Locate a formula for the general term and apply it to get the 100 thterm, given the arithmetic series given the sequence 0.8, 2, 3.2, 4.4, 5.6,.
  2. 4.4, 7.5, 13.7, 16.8,.
  3. 3, 8, 13, 18, 23,.
  4. 3, 7, 11, 15, 19,.
  5. 6, 14, 22, 30, 38,.
  6. 5, 10, 15, 20, 25,.
  7. 2, 4, 6, 8, 10,.
  8. 12,52,92,132,.
  9. 13, 23, 53,83,.
  10. 14,12,54,2,114,. Find the positive odd integer that is 50th
  11. Find the positive even integer that is 50th
  12. Find the 40 th term in the sequence that consists of every other positive odd integer in the following format: 1, 5, 9, 13,.
  13. Find the 40th term in the sequence that consists of every other positive even integer: 1, 5, 9, 13,.
  14. Find the 40th term in the sequence that consists of every other positive even integer: 2, 6, 10, 14,.
  15. 2, 6, 10, 14,. What number is the term 355 in the arithmetic sequence 15, 5, 5, 15, 25,.
  16. What number is the phrase 172 in the arithmetic sequence 4, 4, 12, 20, 28,.
  17. What number is the term 355 in the arithmetic sequence 15, 5, 5, 15, 25,.
  18. 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
  19. 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
  20. This is the problem.
  1. Find a formula for the general term from the terms of an arithmetic sequence given the terms of the series. 1 = 6 and 7 = 42
  2. 1 = 12 and 12= 6
  3. 1 = 19 and 26 = 56
  4. 1 = 9 and 31 = 141
  5. 1 = 16 and 10 = 376
  6. 1 = 54 and 11 = 654. 1 = 6 and 7 = 42
  7. 1= 9 and 31 = 141
  8. 1 = 6 and 7
  1. Find all of the arithmetic means that exist between the two supplied terms. a1=3anda6=17
  2. A1=5anda5=7
  3. A2=4anda8=7
  4. A5=12anda9=72
  5. A5=15anda7=21
  6. A6=4anda11=1

Part B: Arithmetic Series

  1. In light of the general term’s formula, figure out how much the suggested total is. 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. Evaluate. 1160(3n)
  2. 1121(2n)
  3. 1250(4n-3)
  4. 1120(2n+12)
  5. 170 (198n)
  6. 1220(5n)
  7. 160(5212n)
  8. 151(38+14
  9. 1120(1.5n+2.6)
  10. 1175(0.2N1.6)
  11. 1170 (19 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 numbers
  2. The sum of the first 200 positive odd integers
  3. The sum of the first 500 positive odd integers
  4. The sum of the first 50 positive even numbers
  5. The sum of the first 200 positive even integers
  6. The sum of the first 500 positive even integers
  7. The sum of the firstk positive odd integers
  8. The sum of the firstk positive odd integers the sum of the firstk positive even integers
  9. The sum of the firstk positive odd integers
  10. 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
  11. 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. 1,565,450, 2,500,450, k2,
  2. 90,800, k4,230,
  3. 38640, 124,750,
  4. 18,550, k765
  5. 10,578
  6. 20,100,
  7. 2,500,550, k2,
  8. 294 seats, 247 bricks, $794,000, and so on.

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