Given The Arithmetic Sequence An = 3 + 2 N − 1 , What Is The Domain For N? (Question)

Therefore, the domain for n for the given arithmetic sequence an = 3 + 2(n – 1) is all integers where n ≥ 1.

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What is the domain for N in an arithmetic sequence?

The domain in arithmetic (or even geometric) sequence is always all integers where n≥1 or Natural numbers.

What is N in a sequence formula?

What Is n in Arithmetic Sequence Formula? In the arithmetic sequence formula for finding the general term,an=a1+(n−1)d a n = a 1 + ( n − 1 ) d, n refers to the number of terms in the given arithmetic sequence.

What is the sum of the arithmetic sequence 8 14 20 If there are 22 terms?

Summary: The sum of the arithmetic sequence 8, 14, 20 …, if there are 22 terms is 1562.

What is the sum of the arithmetic sequence 6/14 22 If there are 26 terms?

The sum of the arithmetic sequence 6, 14, 22 …, if there are 26 terms is 2756.

What is the sum of a 52 term arithmetic sequence where the first term is 6?

The sum is 4030.

What is the arithmetic formula?

An arithmetic formula is a sequence of numbers that is ordered with a specific pattern. Each successive number is the sum of the previous number and a constant. You can determine the common difference by subtracting each number in the sequence from the number following it.

What is the sum of the arithmetic sequence 8 14 if there are 24 terms?

The sum of the arithmetic sequence 8, 14, 20 …, if there are 24 terms is S24 = 1848.

What is the 7th term of the geometric sequence where a1 − 4096 and a4 64?

What is the 7th term of the geometric sequence where a1 = -4,096 and a4 = 64? Summary: The 7th term of the geometric sequence where a1 = -4,096 and a4 = 64 is -1.

What is the sum of even integers from 10 to 90?

Hence, the sum of even integers from 10 to 90 is 2050.

What is the 22nd term of the arithmetic sequence where a1 8 and a9 56 6 points?

Summary: The 22nd term of the arithmetic sequence where a1 = 8 and a9 = 56 is 134.

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:

  1. 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.

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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.

Given the arithmetic sequence an = 2 − 3(n − 1), what is the domain for n?

The question was posed by maham237 @inMathematics and was seen by 52 people. What is the scope of the number n if we consider the arithmetic sequence an= 2 – 3(n – 1)? A. All integers where n is a positive number (less than or equal to) 1B. All integers where n is a positive number (greater than or equal to) 1C. All integers where n is a positive number (greater than or equal to) 0D is the sum of all integers.

Do you know the better answer?

Asked by admin @inMath on Twitter 38 people have looked at this page. Calculate the total of the following AP: (1) + (1-2/n)+(1-3/n)+.up to nth words (1-1/n)

1 3 2 3 3 3 n 3 formula

Asked by admin @inMath on Twitter 74 people have looked at this page. Use mathematical induction to demonstrate that 1 3 + 2 3 + 3 3 +. + n 3 = n 2 (n + 1) by proving that 1 3 + 2 3 + 3 3 +. + n 3 = n 2 (n + 1). 2 / 4 for everyone.

1 4 2 4 3 4 n 4 formula

Asked by admin @inMath on Twitter 123 people have looked at this page. Find the total of 4-1/n, 4-2/n, and 4-3/n up to the nth term in each of the three terms.

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Evaluate 1 3 2 3 3 3 n 3

Asked by admin @inMath on Twitter 74 people have looked at this page. Use mathematical induction to demonstrate that 1 3 + 2 3 + 3 3 +. + n 3 = n 2 (n + 1) by proving that 1 3 + 2 3 + 3 3 +. + n 3 = n 2 (n + 1). 2 / 4 for everyone.

1 2 3 n 1 8 2n 1 2

Asked by admin @inMath on Twitter 102 people have looked at this page. 1+2+3+.+ nlt; 1/8 (2n +1) square; 1+2+3+.+ nlt; 1+2+3+.+ nlt; 1+2+3+.+ nlt; 1+2+3+.+ nlt;

3 2 1 cube

This question was posed by maham237 @inMathematics and was seen by 58 people. Which rational exponent is equivalent to a cube root in this case? 3/2B. A. 3/2B. 1/2C. 1/4D. 1/3

Sequences Defined by an Explicit Formula

  • Formulate the terms of a sequence that has been specified by an explicit formula. Create a formula for the nth term in a sequence of numbers. The terms of an alternating sequence are to be discovered.

Asequence is a term that may be used to describe an ordered list of numbers. A sequence is a function whose domain is a subset of the counting numbers, and whose output is a sequence of numbers. The amount of hits on the website determines the order in which the pages are displayed. The ellipsis(.) denotes that the sequence will continue forever beyond this point. Each of the numbers in the sequence is referred to as an aterm. The first five words in this sequence are the numbers 2, 4, 8, 16, and 32, respectively.

  • For example, in order to determine the number of hits on the website at the end of the month, it would be necessary to list as many as 31 different phrases.
  • This will save time and effort.
  • Explicit formulae are useful when we want to discover a certain phrase in a series without having to go through all of the preceding terms in the sequence.
  • In our case, each number in the series is twice the preceding number, thus we can create a formula for thentextterm by using powers of two.
  • Finding the next text word in a series may be accomplished by raising the value of thentextpower by 2.
  • There is an explicit formula for this sequence, and it is as follows: We can now answer the question presented at the beginning of this section since we have a formula for the then textterm of the sequence.
  • Identifying the 31 st term of the sequence is necessary for determining the number of hits on the last day of the month.
  • begin = Assuming that the pattern of doubling continues, the firm will get SMS hits on the last day of the month.
  • Consumer interest and competition are not taken into consideration in the calculation of this enormous amount, which makes it rather impractical.

But it does provide a starting point from which the organization might contemplate making strategic business decisions. A table can also be used to show the sequence in a different way. The first five terms of the series, as well as the next textterm in the sequence, are displayed in the table.

n 1 2 3 4 5 n
ntextterm of the sequence,_ 2 4 8 16 32 ^

Graphical representation of the sequence as a collection of different points is provided by graphing the sequence. The graph below shows that the number of hits is increasing at an exponential pace, as can be observed. Exponential functions are formed by this unique series of events. Finally, we may denote this unique sequence by the letters left, dots, and right. It is referred to be an endless sequence when a series continues eternally. Counting numbers are the domain of an infinite sequence, which is represented by the set of counting numbers.

It is referred to as the Afinite sequence since it does not extend endlessly.

A General Note: Sequence

Asequenceis a function whose domain is the set of positive integers and whose domain is the set of positive integers. If the domain of a series contains just the firstn positive integers, then it is called an Afinite sequence. The numbers that make up a sequence are referred to as terms. It is necessary to utilize a variable with a number subscript to represent the terms in a sequence as well as to indicate where a particular phrase is located within the sequence. dots, dots, dots, dots, dots, dots, dots, dots, dots, dots, dots, dots We refer to the first term of the series as the first term of the sequence, the second term of the sequence as the second term of the sequence, and so on.

In an explicit formula, the textterm of a sequence is defined in terms of the term’s position in the sequence.

QA

No. On occasion, it may be advantageous to specify the beginning word as instead of_ while solving a problem. The domain of the function in these issues contains the value 0.

How To: Given an explicit formula, write the firstnterms of a sequence.

  1. Fill in the blanks with the values ofn in the formula. Start with n=1 and work your way up. in order to locate the initial term,_ When looking for the second term,_, usen=2
  2. Continually work in this manner until you have discovered all of the words.

Example: Writing the Terms of a Sequence Defined by an Explicit Formula

Specify the first five terms of the sequence specified by the explicit formula_ =-3n+8 in the space provided.

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Try It

Write the first five terms of the sequence defined by the explicit formula_ =-3n+8 in the form of a sentence.

Investigating Explicit Formulas

Previously, we discovered that sequences are functions with a domain that extends over the positive integers. In addition to piecewise functions, this is true for a variety of different types of functions. Remember that a piecewise function is a function that is defined by a number of different subsections. There might be several distinct formulas used to represent each specific subsection.

How To: Given an explicit formula for a piecewise function, write the firstnterms of a sequence

  1. Decide on the mathematical formula that n=1 pertains to. To locate the first term,_, enter the number n=1. the relevant formula is used
  2. Decide which part of the equation n=2 applies to
  3. When calculating the second term,_, entern=2 in the appropriate formula. Continually work in this manner until you have discovered all of the words.

Example: Writing the Terms of a Sequence Defined by a Piecewise Explicit Formula

Fill in the blanks with the first six terms in the series. Start with text, then text again, then text again, and so on till the conclusion of the document.

Try It

Fill in the blanks with the first six terms in the series. text ntext dfract text ntext nend start 2ntext ntext dfract text text ntext end

Finding an Explicit Formula

So far, we’ve been given an explicit formula and instructed to identify a number of terms in a sequence using that formula. It is possible that the explicit formula for the thentextterm of a sequence will not be provided. Instead, we are provided with a number of words from the sequence. We may then go backwards to find an explicit formula from the first few terms of a series, if this occurs.

It is essential to seek for a pattern in the phrases while trying to develop an explicit formula. Take note that the pattern may include alternating phrases, formulae for numerators, formulas for denominators, exponents, or bases, as well as other elements.

How To: Given the first few terms of a sequence, find an explicit formula for the sequence.

  1. Look for a recurring theme among the terms
  2. If the terms are fractions, search for a distinct pattern in the numerators and denominators
  3. If the terms are decimals, look for a distinct pattern in the numerators and denominators. Consider whether there is a pattern in the signs of the phrases
  4. Formula for in terms ofn should be written. Test your formula using n=1, n=2, and n=3 as inputs.

Example: Writing an Explicit Formula for thenth Term of a Sequence

Create a formula that is explicit for the thentextterm of each sequence.

  1. Dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,-dfrac,- -dfrac,-dfrac,-dfrac125,-dfrac625
  2. -dfrac,-dfrac,-dfrac125,-dfrac625
  3. -dfrac,-dfrac,-dfrac125

Try It

Create an explicit formula for the textterm of the sequence at the end of the series.

Try It

Create an explicit formula for the textterm that occurs at the end of the sequence. left,-dfrac,-dfrac,-dfrac,-dfrac,dotsright}

Try It

Create an explicit formula for the textterm that occurs at the end of the sequence. left, dfrac, 1, e,.right, dfrac, 1, e,.

Investigating Alternating Sequences

Sequences with phrases that switch in sign might appear from time to time. The processes involved in determining the terms of the sequence are the same as they would be if the signs were not alternated. Take a look at the sequence of events that follows. left As you can see in this example, first term is greater than second term; yet, the second term is smaller than third term; and the third term is greater than the fourth term. This pattern will continue indefinitely. It is not necessary to reorder the words in numerical order in order to grasp the sequence.

How To: Given an explicit formula with alternating terms, write the firstnterms of a sequence.

  1. Fill in the blanks with the values ofn in the formula. Start with n=1 and work your way up. in order to locate the initial term,_ In an explicit formula, the sign of the term is determined by the first term, if the first term is negative, and the first term if the first term is positive. When looking for the second term,_, usen=2
  2. Continually work in this manner until you have discovered all of the words.

Example: Writing the Terms of an Alternating Sequence Defined by an Explicit Formula

Fill in the blanks with the first five terms in the series. =dfrac =dfrac =dfrac

QA

Certainly, the power can take on the form of bn+1, bn-1, and so on, but any odd powers will result in a negative term, and any even powers will result in a positive term.

Try It

Write the first five terms of the series in the following format: =dfrac

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