$n^{th}$ term of an A.P. is given by $a_n= a+(n-1)d$. In order to determine the 15th term of the given arithmetic sequence, we relate the **given numbers with the general sequence of A.P.** and Using the $n^{th}$ term formula, we find the 15th term in the given A.P.

Contents

- 1 What is the 15th term in the arithmetic?
- 2 How do you find the term of an arithmetic sequence?
- 3 How do you find the 25th term in a sequence?
- 4 How do you get a15?
- 5 What is the 15th term of Fibonacci sequence?
- 6 What is the sum of the first 15 terms of the arithmetic sequence?
- 7 What is the arithmetic between 10 and 24?
- 8 What is the next term of the arithmetic sequence 3 9 15?
- 9 Introduction to Arithmetic Progressions
- 10 Arithmetic sequences calculator that shows work
- 11 About this calculator
- 12 How to find the nth Term of an Arithmetic Sequence? [Solved]
- 13 Answer: The expression to calculate the n thterm of an arithmetic sequence is a n= a + (n – 1) d.
- 14 Video
- 15 About This Article
- 16 Did this article help you?
- 17 Formulas for Arithmetic Sequences
- 18 Using Explicit Formulas for Arithmetic Sequences
- 18.1 A General Note: Explicit Formula for an Arithmetic Sequence
- 18.2 How To: Given the first several terms for an arithmetic sequence, write an explicit formula.
- 18.3 Example: Writing then th Term Explicit Formula for an Arithmetic Sequence
- 18.4 Try It
- 18.5 A General Note: Recursive Formula for an Arithmetic Sequence
- 18.6 How To: Given an arithmetic sequence, write its recursive formula.
- 18.7 Example: Writing a Recursive Formula for an Arithmetic Sequence
- 18.8 How To: Do we have to subtract the first term from the second term to find the common difference?
- 18.9 Try It

- 19 Find the Number of Terms in an Arithmetic Sequence
- 20 Solving Application Problems with Arithmetic Sequences
- 21 Contribute!

## What is the 15th term in the arithmetic?

From the formula of nth term, the 15th term of the sequence is. a n=arn−1a15=4 (2)15−1=4(2)14=4(16384)=65536. So the 15th 15 t h term of the given geometric sequence is 65536.

## How do you find the term of an arithmetic sequence?

Step 1: The nth term of an arithmetic sequence is given by an = a + (n – 1)d. So, to find the nth term, substitute the given values a = 2 and d = 3 into the formula.

## How do you find the 25th term in a sequence?

An easier way to see this equation is: Y = 4X – 9. To find the 25th term, just plug in 25 for X. Y = 4(25) – 9, making the 25th term in this sequence 91.

## How do you get a15?

If you know the first term of an arithmetic sequence, a1, and the common difference, d, then you can find the nth term, an, using the following rule. Step 3 Use the formula with n = 15 to find the 15th term, a15. an = a1 + (n − 1)d Write the rule. a15 = a1 + (15 − 1)d Substitute n = 15.

## What is the 15th term of Fibonacci sequence?

Therefore, the $ {{15}^{th}} $ term in the Fibonacci sequence of numbers is 377. So, the correct answer is “377”.

## What is the sum of the first 15 terms of the arithmetic sequence?

Thus, the sum of the first fifteen terms in the arithmetic sequence is 975.

## What is the arithmetic 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 next term of the arithmetic sequence 3 9 15?

Solution: A sequence in which the difference between all pairs of consecutive numbers is equal is called an arithmetic progression. The sequence given is 3, 9, 15, 21, 27, … Therefore, the 25th term is 147.

## Introduction to Arithmetic Progressions

Generally speaking, a progression is a sequence or series of numbers in which the numbers are organized in a certain order so that the relationship between the succeeding terms of a series or sequence remains constant. It is feasible to find the n thterm of a series by following a set of steps. There are three different types of progressions in mathematics:

- Arithmetic Progression (AP), Geometric Progression (GP), and Harmonic Progression (HP) are all types of progression.

AP, also known as Arithmetic Sequence, is a sequence or series of integers in which the common difference between two subsequent numbers in the series is always the same. As an illustration: Series 1: 1, 3, 5, 7, 9, and 11. Every pair of successive integers in this sequence has a common difference of 2, which is always true. Series 2: numbers 28, 25, 22, 19, 16, 13,. There is no common difference between any two successive numbers in this series; instead, there is a strict -3 between any two consecutive numbers.

### Terminology and Representation

- Common difference, d = a 2– a 1= a 3– a 2=. = a n– a n – 1
- A n= n thterm of Arithmetic Progression
- S n= Sum of first n elements in the series
- A n= n

### General Form of an AP

Given that ais treated as a first term anddis treated as a common difference, the N thterm of the AP may be calculated using the following formula: As a result, using the above-mentioned procedure to compute the n terms of an AP, the general form of the AP is as follows: Example: The 35th term in the sequence 5, 11, 17, 23,. is to be found. Solution: When looking at the given series,a = 5, d = a 2– a 1= 11 – 5 = 6, and n = 35 are the values. As a result, we must use the following equations to figure out the 35th term: n= a + (n – 1)da n= 5 + (35 – 1) x 6a n= 5 + 34 x 6a n= 209 As a result, the number 209 represents the 35th term.

### Sum of n Terms of Arithmetic Progression

The arithmetic progression sum is calculated using the formula S n= (n/2)

### Derivation of the Formula

Allowing ‘l’ to signify the n thterm of the series and S n to represent the sun of the first n terms of the series a, (a+d), (a+2d),., a+(n-1)d S n = a 1 plus a 2 plus a 3 plus .a n-1 plus a n S n= a + (a + d) + (a + 2d) +. + (l – 2d) + (l – d) + l. + (l – 2d) + (l – d) + l. + (l – 2d) + (l – d) + l. (1) When we write the series in reverse order, we obtain S n= l + (l – d) + (l – 2d) +. + (a + 2d) + (a + d) + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a + d + a … (2) Adding equations (1) and (2) results in equation (2).

+ (a + l) + (a + l) + (a + l) +.

(3) As a result, the formula for calculating the sum of a series is S n= (n/2)(a + l), where an is the first term of the series, l is the last term of the series, and n is the number of terms in the series.

d S n= (n/2)(a + a + (n – 1)d)(a + a + (n – 1)d) S n= (n/2)(2a + (n – 1) x d)(n/2)(2a + (n – 1) x d) Observation: The successive terms in an Arithmetic Progression can alternatively be written as a-3d, a-2d, a-d, a, a+d, a+2d, a+3d, and so on.

### Sample Problems on Arithmetic Progressions

Problem 1: Calculate the sum of the first 35 terms in the sequence 5,11,17,23, and so on. a = 5 in the given series, d = a 2–a in the provided series, and so on. The number 1 equals 11 – 5 = 6, and the number n equals 35. S n= (n/2)(2a + (n – 1) x d)(n/2)(2a + (n – 1) x d) S n= (35/2)(2 x 5 + (35 – 1) x 6)(35/2)(2 x 5 + (35 – 1) x 6) S n= (35/2)(10 + 34 x 6) n= (35/2)(10 + 34 x 6) n= (35/2)(10 + 34 x 6) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) S n= (35/2)(10 + 204) A = 35214/2A = 3745S n= 35214/2A = 3745 Find the sum of a series where the first term of the series is 5 and the last term of the series is 209, and the number of terms in the series is 35, as shown in Problem 2.

Problem 2.

S n= (35/2)(5 + 209) S n= (35/2)(5 + 209) S n= (35/2)(5 + 209) A = 35214/2A = 3745S n= 35214/2A = 3745 Problem 3: A amount of 21 rupees is divided among three brothers, with each of the three pieces of money being in the AP and the sum of their squares being the sum of their squares being 155.

Solution: Assume that the three components of money are (a-d), a, and (a+d), and that the total amount allocated is in AP.

155 divided by two equals 155 Taking the value of ‘a’ into consideration, we obtain 3(7) 2+ 2d.

2= 4d = 2 = 2 The three portions of the money that was dispersed are as follows:a + d = 7 + 2 = 9a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5a = 7a – d = 7 – 2 = 5 As a result, the most significant portion is Rupees 9 million.

## Arithmetic sequences calculator that shows work

This online tool can assist you in determining the first $n$ term of an arithmetic progression as well as the total of the first $n$ terms of the progression. This calculator may also be used to answer even more complex issues than the ones listed above. For example, if $a 5 = 19 $ and $S 7 = 105$, the calculator may calculate the common difference ($d$) between the two numbers. Probably the most significant advantage of this calculator is that it will create all of the work with a thorough explanation.

- + 98 + 99 + 100 =?
- In an arithmetic series, the first term is equal to $frac$, and the common difference is equal to 2.
- An arithmetic series has a common difference of $7$ and its eighth term is equal to $43$, with the common difference being $7$.
- Suppose $a 3 = 12$ and the sum of the first six terms is equal to 42.
- When the initial term of an arithmetic progression is $-12$, and the common difference is $3$, then the progression is complete.

## About this calculator

An arithmetic sequence is a list of integers in which each number is equal to the preceding number plus a constant, as defined by the definition above. The common difference ($d$) is a constant that is used to compare two things. Formulas:The $n$ term of an arithmetic progression may be found using the $color$ formula, where $color$ is the first term and $color$ is the common difference between the first and second terms. These are the formulae for calculating the sum of the first $n$ numbers: $colorleft(2a 1 + (n-1)d right)$ and $colorleft(a 1 + a right)$, respectively.

## How to find the nth Term of an Arithmetic Sequence? [Solved]

In anarithmetic progression, the differences between every two successive terms are the same, resulting in a succession of terms.

## Answer: The expression to calculate the n thterm of an arithmetic sequence is a n= a + (n – 1) d.

Let us have a look at the step-by-step solution. Explanation: Using the following phrase, the thterm of AP may be computed for any given arithmetic sequence. a n= a + (n – 1) d a n= a + (n – 1) d Where,

- This is the first term of the APR
- The common difference is denoted by letter “d”
- The number of terms is denoted by letter “n”
- And the first term of the APR is denoted by the letter “a.”

Let’s look at an example to better comprehend what I’m saying. Example: Figure out what the 25th term is in the above arithmetic sequence of 3, 9, 15, 21, 27,. Solution: In this case, putting these numbers in the formulaa n= a + (n – 1)da 25=3+(25-1) 6a 25=3 + 24 – 1) 6a 25= 3 + 144 – 1) 6a 25= 147Thus. The twenty-fifth phrase in the above sequence is the number 144. We may utilize Cuemath’s Online Arithmetic sequence calculator to identify the arithmetic sequence based on the initial term and the difference between the terms that are common to all of them.

### Hence, the expression to calculate the n thterm of AP if given by a n= a + (n – 1) d.

Documentation Download Documentation Download Documentation An arithmetic sequence is a collection of integers that differ from one another by a fixed amount from one to the next. Consider the following example: the list of even integers. This is an arithmetic sequence since the difference between one number in the list and the next is always 2. It is possible to be requested to discover the very next phrase from a list of terms if you are aware that you are working with an arithmetic sequence.

You can also be asked to fill in a blank if a phrase has been left out. Finally, you could be interested in knowing, for example, the 100th phrase without having to write down all 100 words one by one. You may do any of these tasks with the aid of a few easy steps.

- 1 Determine the common difference between the two sequences. A list of numbers may be given to you with the explanation that the list is an arithmetic sequence, or you may be required to figure it out for yourself. In each scenario, the initial step is the same as it is in the other. Choose the first two terms that appear consecutively in the list. Subtract the first term from the second term to arrive at the answer. It is the outcome of your sequence that is the common difference
- 2 Check to see if the common difference is constant across the board. Finding the common difference between the first two terms does not imply that your list is an arithmetic sequence in the traditional sense. You must ensure that the difference is continuous across the whole list. Subtract two separate consecutive terms from the list to see how much of a difference there is. If the result is consistent for one or two other pairs of words, then you have most likely discovered an arithmetic sequence of terms. Advertisement
- s3 Add the common difference to the last phrase that was supplied. Finding the next term in an arithmetic series is straightforward after you’ve determined the common difference. Simply add the common difference to the final phrase in the list, and you will arrive at the next number in the sequence. Advertisement

- 1 Double-check that you are starting with an arithmetic sequence before proceeding. Sometimes you will have a list of numbers with a missing phrase in the center, and this will be the case. As with the last step, begin by ensuring that your list is an arithmetic sequence. Make a choice between any two consecutive words and calculate the difference between them. Once you’ve done that, compare it to two additional consecutive terms in the list. You can proceed if the differences are the same, in which case you can assume you are working with an arithmetic series. 2 Before the space, add the common difference to the end of the word. This is analogous to appending a phrase to the end of a sequence of words. Locate the phrase in your sequence that comes directly before the gap in question. This is the “last” number that you are familiar with. By multiplying this term by your common difference, you may get the number that should be used to fill in the blank
- 3 To calculate the common difference, subtract it from the phrase that follows the space. Check your response from the other way to be certain that you have the proper answer. An arithmetic sequence should be consistent in both directions, regardless of the direction in which it is performed. If you travel from left to right and add 4, then you would proceed in the opposite direction, from right to left, and do the reverse and remove 4
- 4 is the sum of the two numbers. You should compare your results. The two outcomes that you obtain, whether you add up from the bottom or subtract down from the top, should be identical to one another. If they do, you have discovered the value for the word that was previously unknown. It is your responsibility to ensure that your work is error-free. The arithmetic sequence you have may or may not be correct. Advertisement

- 1 Determine which phrase is the first in the series. Not all sequences begin with the integers 0 or 1 as the first or second numbers. Take a look at the list of numbers you have and identify the first phrase on it. Your beginning position, which can be identified using variables such as a(1), is the following: 2Define your common difference as d in the following way: Find the common difference between the sequences, just like you did previously. The common difference in this working example is 5, which is the most significant. It is the same result if you check with any of the other words in the sequence. This is a common distinction between the algebraic variable d, which we shall observe. 3 Use the explicit formula to solve the problem. In algebra, an explicit formula is a mathematical equation that may be used to determine any term in an arithmetic series without having to write down the entire list of terms in the sequence. An algebraic series can be represented by the explicit formula

- 1 Identify the first phrase in the series by looking at the alphabet. The numerals 0 and 1 are not always the first and second digits of a series. Check the list of numbers you have and see if you can discover the very first phrase on it. Your beginning point, which may be indicated using variables such as a(1), is at the bottom of the page. Assign a letter d to your shared point of difference. The same as previously, find the common difference between the sequences. It is the common difference in this functioning case of 5, which we will use as an example. This result is obtained by checking with other words in the sequence. With regard to the algebraic variable d, we shall take notice of the following common distinction: 3 Use the formula that is explicitly stated. In algebra, an explicit formula is an algebraic equation that may be used to determine any term in an arithmetic series without having to write down the entire list of terms in the sequence. When it comes to algebraic sequences, the explicit formula is

- 4 Fill in the blanks with your information to help us solve the problem. Make use of the explicit formula for your sequence to enter the information that you already know in order to locate the word that you want. Advertisement

- 1, rearrange the explicit formula such that it may be used to solve for additional variables. Several bits of information about an arithmetic sequence may be discovered by employing the explicit formula and some fundamental algebraic operations. As written in its original form, the explicit formula is intended to solve for an integer n and provide you with the nth term in a series of numbers. You may, however, modify this formula algebraically and solve for any of the variables in the equation. 2 Find the first phrase in a series by using the search function. For example, you may know that the 50th term of an arithmetic series is 300, and you may also know that the terms have been growing by 7 (the “common difference”), but you may wish to know what the sequence’s very first term was. To determine your solution, use the improved explicit formula that solves for a1 as previously stated

- Make use of the equation and fill in the blanks with the facts you already know. Because you know that the 50th term is 300, n=50, n-1=49, and a(n)=300 are the values of n. You are also informed that the common difference, denoted by the letter d, is seven. Therefore, the formula is as follows: This works out as well. The series that you have created began at 43 and increased by 7 each time. As a result, it appears as follows: 43,50,57,64,71,78.293,300

- 3 Determine the total length of a sequence. Consider the following scenario: you know the beginning and ending points of an arithmetic series, but you need to know how long it is. Make use of the updated formula

- Consider the following scenario: you know that a specific arithmetic sequence starts at 100 and grows by 13. In addition, you are informed that the ultimate term is 2,856. You can find out the length of the series by putting the terms a1=100, d=13, and a(n)=2856 together. Fill in the blanks with the terms from the formula to get the answer. If you do the math, you will come up with, which equals 212+1, which equals 213. 213 words are included inside a single sequence
- An example of this would be the following: 101-313-126-213-136-139.2843-2856.

Create a new question

- Question How can I determine the first three terms if I only have the tenth and fifteenth terms? Subtract the tenth term from the fifteenth term and divide by five to get D, which is the difference between any two consecutive terms in the series of terms. Calculate the first term by multiplying D by 9 and subtracting that amount from the tenth term
- This is the first term. Question What is the mathematical formula for the numbers 8, 16, 32, 64, and ? This is not an arithmetic sequence in the traditional sense. Research geometric sequences for any formula you’re interested in learning about. Question How do I compute the 5 terms of an arithmetic sequence if the first term is 8 and the final term is 100, and the first term is 8 and the last term is 100? Take 8 away from 100 to get 92. 92 divided by 4 equals (because with five terms there will be four intervals between the first and last term). This gives you the number 23, which is the length of each interval. As a result, the sequence starts with 8 and has a common difference of 23
- Question How can I find out which term in the arithmetic sequence has the value of -38 in it? The common difference (d) is equal to 4 minus 7 = -3. The first term (a) equals 7. The given period (t) equals -38. (n-1)d = t + (a + (n-1)d, or, -38 = 7 + (n-1)-3, is the formula for time. As a result, n=16, which means that -38 is the sixteenth term
- Question The first three terms of 4n+3 are as follows: The first three terms, starting with n = 1, are 7, 11, and 15
- Question In the sequence 1/2, 1, 2, 4, 8, what is the formula for determining the nth term in the sequence? Alexandre Lima’s full name is Alexandre Lima. Community Answer This is a geometric progression in which each phrase is computed by multiplying the previous term by a predetermined constant before proceeding to the next. When using the example, the constant (q) is two since 2 * (1/2) = one, 2 * one = two, and 2 * two equals four. The formula is: a = a1 x q(n-1)
- For example, a = 1/2 x 2 in the example (n-1). For example, the tenth term is written as a(10) = 1/2 x 2(9) = 256. Question What is the best way to discover the 100th term if I only have the first five terms available? Take a look at Method 3 above, particularly Step 3. Question What if you have the common difference and the first term, but you need to know the a specific number is in relation to what nth number? For example, d=-4, a1=35, and 377 is a term number, correct? The formula for the nth term, denoted by the letter a(n), is provided in Method 3 above. Fill in the blanks with your numbers and solve for n
- Question What is the proper way to use the formula? If you want to discover the “nth” term in an arithmetic series, begin with the first term, which is a. (1). In addition, the product of “n-1” and “d” should be considered (the difference between any two consecutive terms). Consider the arithmetic sequences 3, 9, 15, 21, and 27 as an example. Because the difference between successive terms is always six, a(1) = three, and d = six. Consider the following scenario: you wish to locate the seventh word in the series (n = 7). Then a(7) = a(1) + (n-1)(d) = 3 + (6)(6) = 39, and a(7) = a(1) + (n-1)(d) = 39. In this sequence, number 39 corresponds to the seventh word
- Question What is the best way to locate the first three terms? Suppose you have the fourth, fifth, and sixth terms in the series, for example, 6, 8, and ten, respectively. The formula for finding any term in the series is Un (or Ur) = the first term + the term you are attempting to find minus one (for example, if you were trying to find the fifth term, the formula would be 5 -1) x d, where d is the length of the sequence (the common difference). Because you already know some of the terms in the sequence, you can put in the terms you already know into the formula and solve for the first term to get the answer: U(4) = 6 = U(1) + U(2) = U(4) (4-1) 2. The value of the fourth term, U(4), was provided as 6, and the common difference was found to be 2. After being simplified, the formula looks somewhat like this: 6 is equal to U(1) plus 6. The result of removing 6 from both sides is that U(1) equals 0, and you can use this to get any other term in the series using this formula.

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- There are several distinct types of number sequences to choose from. Do not make the mistake of assuming that a list of integers is an arithmetic series. Make sure to verify at least two pairings of words, and ideally three or four, in order to identify the common difference between them.

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## About This Article

Summary of the Article When looking for a term in an arithmetic series, locate the common difference between the first and second numbers by subtracting the first from the second. Verify that the difference is consistent between each number in the series by re-running the preceding equation with the second and third numbers, third and fourth numbers, and so on until the difference is no longer consistent. Once you’ve determined the common difference, all that’s left to do to locate the missing number is to multiply the common difference by the term that came before it in the series.

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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. |

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: |

The fixed number, which is referred to as the common differenceor d-value, is three. We may use this information to replace the explicit rule in the code. As an example, a n= a 1+ (n – 1)d. a n = a 1 + a (n – 1) the value of da n= 5 + (n-1) (3) the number 5 plus 3n – 3a the number 3n + 2a the number 3n + 2 When asked to identify the 37th term in this series, we would compute for a 37 in the manner shown below. the product of 3n and 2a 37 is 3(37) + 2a 37 is 111 + 2a 37 is 113. Exemple No. 2: For sequence B, find a formula that specifies the nth term in the series.

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 TermIt 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. |

## Formulas for Arithmetic Sequences

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

## Using Explicit Formulas for Arithmetic Sequences

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

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

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

It is possible to conceive of anarithmetic sequence as a function on the domain of natural numbers; it is a linear function since it changes at a constant pace over time. It is the constant rate of change, often known as the slope of a function, that is the most commonly seen difference. If we know the slope and vertical intercept of a linear function, we can create the function. # # # # # # # # # # # # # # # # # (n – 1right) For the -intercept of the function, we may take the common difference from the first term in the series and remove it from the result of this operation.

- Considering that the common difference is 50, the series represents a linear function with an associated slope of 50.
- Another method of obtaining the they-intercept is to plot the function on a graph and then determine where a line drawn between two points would intersect the vertical axis.
- The symbols _instead of yandninstead ofx are used when working with sequences.
- When we substitute 50 for the slope and 250 for the vertical intercept, we get the following equation: (50n + 250) = To create an explicit formula for an arithmetic series, we do not need to compute the vertical intercept.

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

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

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

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.

- To discover the common difference between two terms, subtract any phrase from the succeeding term. In the recursive formula for arithmetic sequences, start with the initial term and substitute the common difference

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

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

For the arithmetic series, write out the recursive formula for it. left

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

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

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

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

### Try It

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

## Solving Application Problems with Arithmetic Sequences

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

### Example: Solving Application Problems with Arithmetic Sequences

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

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

### Try It

A lady chooses to go for a 10-minute run every day this week, with the goal of increasing the length of her daily exercise by 4 minutes each week after that. Create a formula to predict the timing of her run after n weeks has passed. In eight weeks, how long will her daily run last on average?

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