How To Write An Arithmetic Sequence? (Question)

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

Contents

What is the example of arithmetic sequence?

What is an arithmetic sequence? An arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term. For example in the arithmetic sequence 3, 9, 15, 21, 27, the common difference is 6. An arithmetic sequence can be known as an arithmetic progression.

What is an arithmetic sequence in math?

Sequences with such patterns are called arithmetic sequences. In an arithmetic sequence, the difference between consecutive terms is always the same. For example, the sequence 3, 5, 7, 9 is arithmetic because the difference between consecutive terms is always two.

What type of sequence is 80 40 20?

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

What is fib 13 )?

The 13th number in the Fibonacci sequence is 144. The sequence from the first to the 13th number is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

What type of sequence is 2 20200 2000?

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by 10 gives the next term. In other words, an=a1⋅rn−1 a n = a 1 ⋅ r n – 1. This is the form of a geometric sequence.

What is the common difference of the arithmetic sequence in 1 5 9 13?

For example, in the arithmetic sequence 1, 5, 9, 13, 17, …, the difference is always 4.

What sequence is formed by the reciprocals of the arithmetic sequence?

A harmonic progression is a sequence of real numbers formed by taking the reciprocals of an arithmetic progression. Equivalently, it is a sequence of real numbers such that any term in the sequence is the harmonic mean of its two neighbors.

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:

Algebra II Recipe: Arithmetic Sequences

  1. There are no variations! As well as the fact that there are “n” of them The number 2S equals the sum of the numbers (2a + (n1)d) and the number 2S equals two times the number two. Simply divide the result by two to obtain: The function S = (n/2) (2a + (n1)d) is defined as follows: What we’ve come up with is this:
  • Arithmetic sequences are formed when there is a common difference of 5 between the numbers 4, 9, 14, 19, 24, and so on. Arithmetic sequences are formed when there is a common difference of -3 between the numbers 17, 14, 11, 8, 5, and so on.
  1. An equation that allows you to locate any term in a series is known as a rule.

In the case of an Arithmetic Sequence, the following rule applies: a 1+ (n – 1)d

  1. A n is the n thterm in the series
  2. A 1 is the first term of the sequence
  3. N is the number of terms in the sequence
  4. D is the common difference
  5. And n is the number of terms in the sequence To write the rule, just the a 1 and d values should be used.

C. Creating a Rule When You Only Have the Arithmetic Sequence As Information

  1. Calculate the values of a 1 and d
  2. Substitute the values of a 1 and d into the equation a n= a 1+ (n – 1)d
  3. Reduce the complexity of the equation
D. Writing a Rule When You Know Some Term In the Arithmetic Sequence and the Common Difference.
  1. 1 is obtained by putting the provided information into a n= an n+1 + (n – 1)d. Only the a 1 and d values should be substituted into a n= a 1+ (n – 1)d
  2. Reduce the complexity of the equation
E. Writing a Rule When You Only Know Two Terms in the Arithmetic Sequence.
  • N= a 1+ (n – 1)d is substituted into Eq. 1 for the biggest n, and n= a 1+ (n – 1)d is substituted into Eq. 2 for the smallest n, and so on.
  1. Simplify each equation
  2. Subtract the equations (Eq. 1-Eq. 2) to arrive at d
  3. Simplify each equation again. In order to determine a 1, substitute the value of d intoEq. 2 (the “smallest equation”). Substitute the values of a 1 and d into the equation a n= a 1+ (n – 1)d
  4. Simplify the expression

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.

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.

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. 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

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. When determining the number of terms in a finite arithmetic sequence, explicit formulae can be employed to aid in the calculation of the result. 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 important tasks.

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?

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Arithmetic Sequences

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

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

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

Arithmetic 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 Sequence – Formula, Meaning, Examples

When you have a succession of integers where the differences between every two subsequent numbers are the same, you have an arithmetic sequence. Let us take a moment to review what a sequence is. A sequence is a set of integers that are arranged in a certain manner. An arithmetic sequence is defined as follows: 1, 6, 11, 16,. is an arithmetic sequence because it follows a pattern in which each number is acquired by adding 5 to the phrase before it. There are two arithmetic sequence formulae available.

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

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

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

What is an Arithmetic Sequence?

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

Arithmetic Sequences Example

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

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

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

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

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

First Term of Arithmetic Sequence

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

Common Difference of Arithmetic Sequence

The addition of a fixed number to each preceding term in an arithmetic series, with one exception (the first term), has previously been demonstrated in prior sections.

The “fixed number” in this case is referred to as the “common difference,” and it is symbolized by the letter d. The formula for the common difference isd = a – an1.

Nth Term of Arithmetic Sequence Formula

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

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

Arithmetic Sequence Recursive Formula

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

Example: If a19 = -72 and d = 7, find the value of a21 in an arithmetic sequence.

a21 = a20 + d = -65 + 7 = -58; a21 = a20 + d = -65 + 7 = -58; a21 = a20 + d = -65 + 7 = -58; As a result, the value of a21 is -58.

Sum of Arithmetic sequence Formula

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

  • The following is true: When the n thterm is unknown, Sn= n/2
  • When the n thterm is known, Sn= n/2

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

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

Sum of Arithmetic Sequence Proof

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

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

By putting a = a1 + (n – 1)d into the previous formula, we getSn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)S As a result, we have obtained both the sum of the arithmetic series and the sum of the logical sequence.

Arithmetic Sequence Formulas

Consider the following arithmetic sequence: a1 is the first term, and d is the common difference between the first and second terms. Then Sn = a1 + (a1 + d) + (a1 + 2d) +. + an is the sum of the first ‘n’ elements in the sequence. Consider the following example: (1)Write the same amount from right to left from right to left (i.e., from the n thterm to the first term). (an – d) + (an – 2d) +. + a1. Sn = a plus (an – d) plus (an – 2d) plus. +a1. When you combine (1) and (2), all words beginning with the letter “d” are eliminated.

+ (a1 + an) 2Sn = (a1 + an) + (a1 + an) +.

The number n equals two.

  • Consider an arithmetic sequence in which the first term is a1 and the common difference is d. The sum of the first ‘n’ terms of the series is given bySn = a1 + (a1 + d) + (a1 + 2d) +. + an, where Sn = a1 + (a1 + d) + (a1 + 2d) +. + an (1)Let us write the same total from right to left in the same manner (i.e., from the n thterm to the first term). Sn = (an – d) + (an – 2d) +. + a1. (2)By combining (1) and (2), all phrases beginning with the letter ‘d’ are canceled. 2Sn = (a1 + an) + (a1 + an) + (a1 + an) + (a1 + an) +. + (a1 + an) 2Sn = n (a1 + an) = n (a1 + an). Sn =/2 is a mathematical formula. By putting a = a1 + (n – 1)d into the previous formula, we obtainSn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)Sn = n/2(or)S As a result, we have obtained both the sum of the arithmetic series and the sum of the geometric sequence.

Difference Between Arithmetic and Geometric Sequence

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

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

Notes on the Arithmetic Sequence that are very important

  • Arithmetic sequences have the same difference between every two subsequent numbers
  • This is known as the difference between two consecutive numbers. The common difference of an arithmetic sequence a1, a2, and a3 is d = a2 – a1 = a3 – a2 =
  • The common difference of an arithmetic sequence a1, a2, and a3 is d = a2 – a1 = a3 – a2 =
  • It is an= a1 + (n1)d for the n-th term of an integer arithmetic sequence. It is equal to n/2 when the sum of the first n terms of an arithmetic sequence is calculated. Positive, negative, or zero can be used to represent the common difference of arithmetic sequences.

The difference between every two consecutive integers in an arithmetic series is the same; If we consider an arithmetic sequence of the numbers a1, a2, and a3, then the common difference is denoted by the symbol, d = a2 – A1 = a3 – A2 =, and the common difference of an arithmetic sequence of the numbers a1, a2, and a3 is denoted by the symbol, d = a1 = a3 – A2 = In an arithmetic series, the n th term is an= a1 + (n1)d.

In an arithmetic series, the sum of the first n terms is Sn = n/2. Positive, negative, or zero can be used to describe the common difference of arithmetic sequences.

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

Solved Examples on Arithmetic Sequence

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

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FAQs on Arithmetic sequence

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What are Arithmetic Sequence Formulas?

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

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

How to Find An Arithmetic Sequence?

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

What is the n thterm of an Arithmetic Sequence Formula?

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

What is the Sum of an Arithmetic Sequence Formula?

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

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

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

How to Find n in Arithmetic sequence?

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

How To Find the First Term in Arithmetic sequence?

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

What is the Difference Between Arithmetic Sequence and Arithmetic Series?

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

What are the Types of Sequences?

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

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

What are the Applications of Arithmetic Sequence?

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

How to Find the n thTerm in Arithmetic Sequence?

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

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

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

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

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

Arithmetic Sequences and Series

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

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

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

Example 1

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

To determine the 100th term, use the following equation: a100=3(100)+4=304 Answer_an=3n+4;a100=304 It is possible that the common difference of an arithmetic series be negative.

Example 2

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

Example 3

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

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

Example 4

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

an=a1+(n−1)d=−15+(n−1)⋅7=−15+7n−7=−22+7n Answer_an=7n−22 Take a look at this! Identify the general term of the above arithmetic sequence and use that equation to determine the series’s 100th term. For example: 32,2,52,3,72,… Answer_an=12n+1;a100=51

Arithmetic Series

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

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

2.:Sn=n(a1+an) 2 Calculate the sum of the first 100 terms of the sequence defined byan=2n1 by using this formula.

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

Example 5

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

Example 6

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

Example 7

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.

Take a look at this! Calculate the sum of the first 60 terms of the following sequence of numbers: 5, 0, 5, 10, 15,. are all possible combinations. Answer_S60=−8,550

Key Takeaways

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

Topic Exercises

  1. Given the first term and common difference of an arithmetic series, write the first five terms of the sequence. Calculate the general term for the following numbers: a1=5
  2. D=3
  3. A1=12
  4. D=2
  5. A1=15
  6. D=5
  7. A1=7
  8. D=4
  9. D=1
  10. A1=23
  11. D=13
  12. A 1=1
  13. D=12
  14. A1=54
  15. D=14
  16. A1=1.8
  17. D=0.6
  18. A1=4.3
  19. D=2.1
  1. Find a formula for the general term based on the arithmetic sequence and apply it to get the 100 th term based on the series. 0.8, 2, 3.2, 4.4, 5.6,.
  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. Calculate a formula for the general term based on the terms of an arithmetic sequence: a1=6anda7=42
  2. A1=12anda12=6
  3. A1=19anda26=56
  4. A1=9anda31=141
  5. A1=16anda10=376
  6. A1=54anda11=654
  7. A3=6anda26=40
  8. A3=16andananda15=
  1. Find all possible arithmetic means between the given terms: a1=3anda6=17
  2. A1=5anda5=7
  3. A2=4anda8=7
  4. A5=12anda9=72
  5. A5=15anda7=21
  6. A6=4anda11=1
  7. A7=4anda11=1

Part B: Arithmetic Series

  1. Make a calculation for the provided total based on the formula for the general term an=3n+5
  2. S100
  3. An=5n11
  4. An=12n
  5. S70
  6. An=132n
  7. S120
  8. An=12n34
  9. S20
  10. An=n35
  11. S150
  12. An=455n
  13. S65
  14. An=2n48
  15. S95
  16. An=4.41.6n
  17. S75
  18. An=6.5n3.3
  19. S67
  20. An=3n+5
  1. Consider the following values: n=1160(3n)
  2. N=1121(2n)
  3. N=1250(4n3)
  4. N=1120(2n+12)
  5. N=170(198n)
  6. N=1220(5n)
  7. N=160(5212n)
  8. N=151(38n+14)
  9. N=1120(1.5n2.6)
  10. N=1175(0.2n1.6)
  11. The total of all 200 positive integers is found by counting them up. To solve this problem, find the sum of the first 400 positive integers.
  1. The generic term for a sequence of positive odd integers is denoted byan=2n1 and is defined as follows: Furthermore, the generic phrase for a sequence of positive even integers is denoted by the number an=2n. Look for the following: The sum of the first 50 positive odd integers
  2. The sum of the first 200 positive odd integers
  3. The sum of the first 50 positive even integers
  4. The sum of the first 200 positive even integers
  5. The sum of the first 100 positive even integers
  6. The sum of the firstk positive odd integers
  7. The sum of the firstk positive odd integers the sum of the firstk positive even integers
  8. The sum of the firstk positive odd integers
  9. There are eight seats in the front row of a tiny theater, which is the standard configuration. Following that, each row contains three additional seats than the one before it. How many total seats are there in the theater if there are 12 rows of seats? In an outdoor amphitheater, the first row of seating comprises 42 seats, the second row contains 44 seats, the third row contains 46 seats, and so on and so forth. When there are 22 rows, how many people can fit in the theater’s entire seating capacity? The number of bricks in a triangle stack are as follows: 37 bricks on the bottom row, 34 bricks on the second row and so on, ending with one brick on the top row. What is the total number of bricks in the stack
  10. Each succeeding row of a triangle stack of bricks contains one fewer brick, until there is just one brick remaining on the top of the stack. Given a total of 210 bricks in the stack, how many rows does the stack have? A wage contract with a 10-year term pays $65,000 in the first year, with a $3,200 raise for each consecutive year after. Calculate the entire salary obligation over a ten-year period (see Figure 1). In accordance with the hour, a clock tower knocks its bell a specified number of times. The clock strikes once at one o’clock, twice at two o’clock, and so on until twelve o’clock. A day’s worth of time is represented by the number of times the clock tower’s bell rings.

Part C: Discussion Board

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

Answers

  1. 5, 8, 11, 14, 17
  2. An=3n+2
  3. 15, 10, 5, 0, 0
  4. An=205n
  5. 12,32,52,72,92
  6. An=n12
  7. 1,12, 0,12, 1
  8. An=3212n
  9. 1.8, 2.4, 3, 3.6, 4.2
  10. An=0.6n+1.2
  11. An=6n3
  12. A100=597
  13. An=14n
  14. A100=399
  15. An=5n
  16. A100=500
  17. An=2n32
  1. 2,450, 90, 7,800, 4,230, 38,640, 124,750, 18,550, 765, 10,000, 20,100, 2,500, 2,550, K2, 294 seats, 247 bricks, $794,000, and

Sequences as Functions – Explicit Form- MathBitsNotebook(A1

We saw inSequences – Basic Information, that sequences can be expressed in various forms.This page will look at one of those forms, the explicit form.Certain sequences (not all) can be defined (expressed) as an “explicit” formula.An explicit formulawill create a sequence usingn, the numberlocation of each term. If you can find an explicit formula for a sequence, you will be able to quickly and easily find any term in the sequence simply by replacingnwith the number of the term you seek.Anexplicit formuladesignates thenthterm of the sequence,as an expression ofn(wheren= the term’s location). It defines the sequence as a formula in terms ofn. It may be written in either subscript notationan, or in functional notation,f(n).Sequence: . Find an explicit formula. This example is anarithmetic sequence(the same number, 5, is added to each term to get tothe next term).

Term Number Term Subscript Notation Function Notation
1 10 a1 f(1)
2 15 a2 f(2)
3 20 a3 f(3)
4 25 a4 f(4)
5 30 a5 f(5)
6 35 a6 f(6)
n a n f(n)

Explicit Formula: in subscript notation_an= 5 n+ 5 in function notation:f(n) = 5 n+ 5

This sequence is graphed in the first quadrant. Remember that the domain consists ofthe natural numbers, and the range consists of the terms of the sequence.Notice that this sequence has alinear appearance.The rate of change between each of the points is “5over 1”.While thenvalue increases by a constant value of one, thef(n) value increases by a constant value of 5(for this graph).Willarithmetic sequences be linear functions?

It is easy to see that the explicit formula works once you are giventhe formula. Unfortunately, it is not always easy to come up withexplicitformulas,when all you have is a list of the terms.If your sequence is arithmetic, it will help if you look at the pattern of what is happening in the sequence.
Explicit formula:f(n) = 10 + 5(n- 1) If you compare the term number with how many times the common difference, 5, is added, you will see a pattern for an explicit formula:

For the purpose of summarizing the process of developing an explicit formula for an arithmetic series, consider the following: 1st, determine whether the sequence is arithmetic (does each phrase add or remove the same amount from the previous term) 2.Identify the common point of disagreement. (It is the number you add or subtract from the total.) 3.Create an explicit formula using the pattern of the first term added to the product of the common difference and one less than the term number and one more than the term number.

a n=a1+d(n- 1) a n= then thterm in the sequencea1= the first term in the sequencen= the term numberd= the common difference.
first term = 10, common difference = 5explicit formula:a n= 10 + 5(n- 1)= 10 + 5 n- 5 = 5 + 5 nor 5 n +5

Using the explicit formula, identify the 100 th term in this series of numbers. In the explicit formula, 100 has been substituted for: f(n) = 10 + 5 = f(n) (n- 1) 100 = 10 + 5(100 – 1) = 10 + 5(99) = 10 + 495 =505 100 + 5(99) = 10 + 495 =505 THE ANSWERSequence is:. Look for a formula that is explicit. The following is an example of an ageometric sequence (the same number, 2, is multiplied times each term to get tothe next term).

Term Number Term Subscript Notation Function Notation
1 3 a1 f(1)
2 6 a2 f(2)
3 12 a3 f(3)
4 24 a4 f(4)
5 48 a5 f(5)
6 96 a6 f(6)
n a n f(n)

Explicit Formula:in subscript notation_an= 3 (2)n -1 in function notation:f(n) = 3 (2)n -1

Notice that this sequence has anexponential appearance.It may be the case with geometric sequences that the graph will increase (or decrease). The rate of change will increase (or decrease) as the value ofnincreases (it is not constant). Will such geometric sequences be exponential functions?

Again, it is easy to see that a given explicit formula works. The problem is coming up with a formula when all you are given is a list of the terms.If your sequence is geometric, it will help if you look at the pattern of what is happening in the sequence, in a manner similar to what we examined in the arithmetic sequence.
Explicit formula:f(n) = 32n -1 If you compare the term number with the powers ofthe common difference, 2,you will see a pattern for an explicit formula:

This is a brief summary of the steps involved in constructing an explicit formula for a geometric sequence: 1.Check to see if the sequence is geometric (i.e., do you multiply or divide by the same number from one term to the next?) 2. Calculate the common ratio. (The result of multiplying or dividing a number.) Develop an explicit formula utilizing the pattern of the first term multiplied by the common ratio raised to a power of one less than the term number, and then evaluate the expression.

a n= then thterm in the sequencea1= the first term in the sequencen= the term numberr= the common ratio
first term = 3, common ratio = 2explicit formula:a n= 32n -1

Now that you have the exact formula, you must determine the 9 thterm in this series of numbers.

In the explicit formula, the number 9 has been substituted: f(n) = 32n-1 f(9) = 32 9-1= 32 8= 3256 = 768 f(n) = 32n-1 f(9) = 32 9-1= 32 8= 3256 = 768 f(n) = 32n-1 f(9) = 32 9-1= 32 8= 3256 = 768 Sequence: This example is neither an arithmetic nor a geometric sequence in the traditional sense.

Seeing the pattern for an explicit formula foran arithmetic sequence or a geometric sequence will be easy ascompared to findingexplicit formulas for sequences that do not fall into these categories.The sequence shown in this example isa famous sequence called theFibonacci sequence.
The Fibonacci sequence is famous as being seen in nature (leaf arrangements, bracts of pine cones, scales of pine cones, sunflowers, flower petals, Nautilus shells, grains of wheat, coniferous trees, bee hives, and even single cells). It isreferred to as Nature’s numbering system. The Fibonacci sequence is alsoassociated with the golden ratio (1.61803), which can be seen in the ratio of two successive terms of the Fibonacci sequence (as the Fibonacci numbers grow).
Is there apattern in the Fibonacci sequence?Yes. After the first two terms, each term is thesum of the previous two terms.Is there an explicit formula forthe Fibonacci sequence?Yes.

Explicit Formula: (Don’t panic!You will not be asked to find explicit formulas of this difficulty level.But you may be asked to “use” a more difficult given formula.)
What about the graph of the Fibonacci sequence?As seen at the right, when graphed, the Fibonacci sequence takes on the appearance of an exponential graph. While it is not truly exponential, the Fibonacci sequence can be “modeled” with an exponential function. With the sequence’s connection to the golden ratio, it canbe “modeled” by an exponential function with 1.6 as the base,f(x) = 1.6x. (This is a “model”, not an exact formula match.)

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