# Arithmetic Sequence Nth Term

Arithmetic and Geometric Sequence, Sum, Nth Term, Cheat Sheet - Foldable Create a foldable or just pass out the Series and Sequence cheat sheet, the choice is yours. The explicit formula. In our example, d = ____. All the linear sequences content remains (see my previous post about methods for finding an nth term). If a1 is the first term of an arithmetic sequence, an the nth term, d is the common difference, a formula for finding the value of the nth term of an arithmetic sequence is: an = a1 + (n - 1)d. See explanation. Arithmetic Sequences An arithmetic sequence is a sequence where each term is found by adding a constant to the previous term. Similarly, in a geometric sequence, the nth term is given by 1an=a1•rn−1 , where r is the common ratio. Use these formulas to determine the indicated term in the given sequence. for finding the nth term. arithmetic (13) arithmetic mental (1) arithmetic sequences (2) arrangements (1) art (11) averages (22) bar charts (2) bearings (2) being systematic (5) best buys (1) bidmas (5) big numbers (1) bisectors (1) boxplots (5) brackets (2) calculator use (2) chanting (1) circle equation (1) circle mensuration (15) circle theorems (8) coins (1) combinations (1) compactness (1) compound interest (1). 1, find the 16th term. There are many problems we can solve if we keep in mind that the nth term of an arithmetic sequence can be written in the following way: a n = a 1 +(n - 1)d Where a 1 is the first term, and d is the common difference. y-intercept of the line. The 5th term of an arithmetic sequence is 8. Find a formula for the nth term. Write down the first 5 terms of the sequence. Find the common difference between the terms. Arithmetic Sequences Given the explicit formula for an arithmetic sequence find the common difference and the 52nd term. ""7 10 13 16 19 "(a) Find the 10th term in this number sequence. Prove that the sum Sn of n terms of an Arithmetic Progress (A. Learn about it The following is a series of slides and videos that will help you understand, learn about and review this sub-topic. The terms arithmetic progression and geometric progression are now used in both the specification and sample assessment materials - terms I normally don't introduce to students until Year 12. - Find an equation for the nth term of the arithmetic sequence. 1 n Find a 27 10) a n = 11 8 + 1 2 n Find a 23 Given the first term and the common difference of an arithmetic sequence find the first five terms. This Bingo game gives students practice of generating the first 4 terms of an arithmetic sequence from its nth term or vice versa. The first four terms in an arithmetic sequence are x+y, x-y, xy, x/y, in that order. Arithmetico-geometric sequences arise in various applications, such as the computation of expected values in probability theory. We'll review some examples of how to use this formula to find a term, the term number, or the common difference. Arithmetic Sequences and Sums Sequence. 3, 6, 12, 24, 48, … Write an equation for this arithmetic sequence and find the. Arithmetic Sequence Problems 1. OR An Arithmetic Sequence is a sequence whose difference between consecutive terms is a constant value. x 2 = a + d. Let’s have an example to help you understand the concept better. In your sequence, a = 5, and d = -3. In an arithmetic sequence like this:. Definition An arithmetic sequence is a list of numbers whose terms all differ by the same non-zero number. Sequences Objectives Find the nth term of a sequence. If a sequence is formed by adding (or subtracting) the same number each time to get the next term, it's called an arithmetic sequence. In an arithmetic progression, it is possible to figure out the sum of n terms manually, using Sum of n terms in Arithmetic Progression =. Find the nth term of the following arithmetic sequence 5, 8, 11, 14, 17, Then find the 100th term. 8, 6, 4, 2, -Find the first six terms of the sequence. Add to that the product of "n-1" and "d" (the difference between any two consecutive terms). If we multiply, it is a geometric sequence. A sequence is called an arithmetic progression (abbreviated A. Arithmetic and geometric progressions are particular types of sequences of numbers which occur frequently in business calculations. The "Nth" term in a mathematical equation is used to represent an unknown position in a geometrical sequence. Similarly, in a geometric sequence, the nth term is given by 1an=a1•rn−1 , where r is the common ratio. 3) An arithmetic sequence is as follows: 2, 24, 46… a. consisting of m terms, then the nth term from the end will be = ar m-n. How do you use a arithmetic sequence to find the nth term? The difference between successive terms in an arithmetic sequence is a constant. You need to provide the first term of the sequence (\(a_1\)), the difference between two consecutive values of the sequence (\(d\)), and the number of steps (\(n\)). Reviewing common difference, extending sequences, finding the nth term, finding a specific term in an arithmetic sequence, recursive formula, explicit formula. 3, 6, 12, 24, 48, … Write an equation for this arithmetic sequence and find the. This constant difference makes your sequence an arithmetic sequence. Sequences vs series, arithmetic sequences have a common difference d, how to find the nth term of an arithmetic sequence. 6, 11, 16, 21, 26 Find an expression, in terms of n, for the nth term of the sequence. What is the first term? Let be the common difference, and let be the second term. where r is the common ratio. Similarly, in a geometric sequence, the nth term is given by 1an=a1•rn−1 , where r is the common ratio. Learn Java by examples. An arithmetic sequence is a sequence where you can add the same number to each term to get to the next one. Nth term of an arithmetic and geometric sequence. The term before the nth term How can you determine if a sequence is arithmetic The common difference, d, must be the same for each successive term in the sequence. If a is the first term, r is the common ratio of a finite G. Writing out or numbers takes too. You would figure out the formula an = a1+(n-1)d-10 where an is your y-value, a1 is your first term in a number sequence (your x. An example is shown where a 1 and d may entered as integers, decimals or fractions and n must be a positive integer. 3) finding the_nth_term 1. - Find an equation for the nth term of the arithmetic sequence. We call this constant value the common difference (\(d\)). aa n dn 1 (1) where an is the term that you are looking for, a1 is the first term in the sequence and d is the common difference of the sequence. 2 Analyzing Arithmetic Sequences and Series 419 Writing a Rule for the nth Term Write a rule for the nth term of each sequence. Algebra 1 - Arithmetic sequences Arithmetic sequences is a part of syllabus in algebra 1 (second math course), which finds application in many algebra questions including algebra word problems. Since 2 and 3 are constants, if we let a be the first term of the sequence and d be the constant difference, then the formula that will describe the nth term of the sequence is. Arithmetic Sequences. Use these formulas to determine the indicated term in the given sequence. The first term of the sequence can be written as u 1. The following diagrams give an arithmetic sequence and the formula to find the n th term. Arithmetic Progressions. tn is called the _____. nth Term Calculator Our online nth term calculator helps you to find the nth position of the sequence instantly. Similarly, in a geometric sequence, the nth term is given by 1an=a1•rn−1 , where r is the common ratio. For instance, to find the general formula of an arithmetic sequence where a4 = –23 and a22 = 40, follow these steps: Find the common difference. What is the common difference for Livy’s sequence? Show or explain how you got your answer. ) if the difference of the term and the preceding term is always same or constant. In mathematics, an arithmetico–geometric sequence is the result of the term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. Give a recursive formula for the sequence. This constant difference is called common difference. Students are required to be able to identify, generate and generalise arithmetic sequences. Since these are square numbers then the nth term of this sequence is n^2. What is the formula for the nth term of an arithmetic sequence? a n = a 1 + (n-1)d, where a 1 is the first term and d is the common difference. Now we know that the second term is 37. [2] IQ (a) Write an expression, in terms of n, for the nth term of this sequence. If the common difference is 3, what is the 1st term?. Learn faster with spaced repetition. Here it is 6. An arithmetic sequence is a sequence in which the difference between each consecutive term is constant. Example 1: Find the 27 th term of the arithmetic sequence 5 , 8 , 11 , 54 ,. Then use the formula for the sum of an arithmetic sequence. n n n is the number of terms in the sequence. Solution: Yes, it is an arithmetic progression. You know the second term: x - y. OR An Arithmetic Sequence is a sequence whose difference between consecutive terms is a constant value. The formula is ONLY for arithmetic sequences where d remains constant. Example 1: 0,6, 12, 18, 24, each term of the sequence is obtained by adding 6 to the preceding term. Figure 2 shows the graph of the arithmetic sequence and its trend line denoted by the dashed line. Prove that the sum Sn of n terms of an Arithmetic Progress (A. a 15 = 272, d = 20. In an arithmetic sequence, the third term is 10 and the fifth term is 16. Let me know if you need more help with your maths on the forum. Download the solution to the activity sheet here. is an arithmetic sequence. In this Chapter we learn about SequencesSequence is any group of numbers with some pattern. Data given: 11th term and 31st term of the same arithmetic sequence. So my goal here is to figure out which of these sequences are arithmetic sequences. What is the fifth term? Well, you know the first term: x + y. n2 + 5 Look for a pattern and then predict the general term, or nth term, an, of the sequence. Program for N-th term of Arithmetic Progression series Given first term (a), common difference (d) and a integer N of the Arithmetic Progression series, the task is to find N th term of the series. 160 + 80 + 40 + … , n=6 13. Arithmetic Progressions. The first term is 20, the second is 40, and the nth term is given by the formula x = 20 times n. 1 - Enter the first term A1 in the sequence, the common difference d and n the number of terms in the sum then press enter. Arithmetic Sequences and Partial Sums Example 3: The fourth term of an arithmetic sequence is 20 and the 13 th term is 65. I am struggling to find anything that gives a good real world example of an arithmetic sequence or why we use the nth term calculation in real life. Formula The nth term of an arithmetic sequence can be found using a n a 1 n 1 d where a 1 is the first term and d is the common difference. a1 is the first term, An is a specific term(so if we're looking for the 9th term, then it would be a9), d is the common difference(the d refers to the fact that the difference between two successive terms yields the constant value that was added). A series of terms is known as a HP series when their reciprocals are in arithmetic progression. Usually, the formula for the nth term of an arithmetic sequence whose first term is a 1 and whose common difference is d is displayed below. 3) a , d 4) a. An arithmetic sequence is a sequence such that each successive term is obtained from the previous term by addition or subtraction of a fixed number called a common difference. See Prentice Hall's Mathematics Offerings at: http://www. NAME:_____ Use the formulas provided to you to complete the following. Substitute a ,d values in n th term. 1) -19, Infinite Algebra 2 - Arithmetic Sequences: Finding the nth Term. Leonardo Fibonacci, who was born in the 12th century, studied a sequence of numbers with a different type of rule for determining the next number in a sequence. What is the first term? Let be the common difference, and let be the second term. A sequence is called an arithmetic progression (abbreviated A. Here's what we use this for: The nth term is given by a formula. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. Arithmetic Sequences. If we wanted to find the 22nd term of that sequence, which is referred to as the Nth term, we would find the relationship between each number, which is. Section 1 Arithmetic Progression An arithmetic progression is a list of numbers where the di erence between successive numbers is constant. Basically we need to find two things, the first term of the sequence and the common difference, d. What does the graph of an arithmetic sequence look like? What does that mean about the common difference? Example 2: Write the first five terms of the sequence. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. com/math Selected images used courtesy Texas Instruments Incorporated. 1 n Find a 27 10) a n = 11 8 + 1 2 n Find a 23 Given the first term and the common difference of an arithmetic sequence find the first five terms. But doing it the other way around is a struggle. For example, the sequence 2, 4, 6, 8, … 2, 4, 6, 8, \dots 2, 4, 6, 8, … is an arithmetic sequence with the common difference 2 2 2. Arithmetic Progressions. Program for N-th term of Arithmetic Progression series Given first term (a), common difference (d) and a integer N of the Arithmetic Progression series, the task is to find N th term of the series. For example, 1/3, 2/3, 1, 4/3 is arithmetic, since you obtain every term by adding 1/3 to the previous term. a 1 = 5, d = 5, n = 25. Sum of Arithmetic Sequence Formula. -- Sum of first n terms. The sequences can also be calculated by recurrence, for that, it is necessary to use the the calculator of sequences defined by recurrence. The second term of an arithmetic sequence is 7, the sum of the first four terms in the arithmetic sequence Write the rule for the nth term of an arithmetic sequence in which a10=46 and d=3 The third and thirtieth term of an arithmetic sequence are 4 and 85 write an explicit rule for this sequence. Find a formula for the nth term. So already, just from the first three terms, we can see that this is not an arithmetic sequence. Posts about sequences written by corbettmaths. You can derive an arithmetic sequence formula that allows you to calculate the nth term in any sequence. Arithmetic Progressions What is the 6 t h 6 ^{th} 6 t h term of. How to recognize, create, and describe an arithmetic sequence (also called an arithmetic progression) using closed and recursive definitions. P, we usually denote the first term by a, the common difference by d and the nth term by tn. Alternative Formula for the Sum of an Arithmetic Sequence 4. By determining the value of the nth term, one can identify the values of terms that appear much later in the sequence without needing to rely on numerous repeated computations. Then use the formula for a Subscript nan to find a 20a20 , the 2020th term of Write a formula for the general term (the nth term) of the arithmetic sequence. There is a formula for finding the nth term of an arithmetic sequence: t n = a + (n-1)d where tn represents the nth term a represents the first term n represents the number of terms d represents the common difference between the terms. d d d is the common difference between any two consecutive terms (arithmetic sequences only). If a 1 is the first term of an arithmetic sequence and d. The fifth is 10. Find the 35th term in the sequence. a is first term and d is common difference. What is the difference between each term in an arithmetic sequence, if the first term of the sequence is -6 and the 12th term is 126? 3. The Fibonacci calculator uses the following generalized formula for determining the n. Let's verify the nth term of our sequence:. Sequences of numbers that follow a pattern of adding a fixed number from one term to the next are called arithmetic sequences. a n = b n /c n. Alternative Formula for the Sum of an Arithmetic Sequence 4. y-intercept of the line. Multiple worksheets. If we add a number to get from one element to the next, we call it an arithmetic sequence. } We will represent the nth term using. Its first term is 1 and the common differnece is 10. The nth term can be explained as the expression which helps us to find out the term which is in nth position of a sequence or progression. In an arithmetic sequence like this:. Arithmetic and geometric progressions are particular types of sequences of numbers which occur frequently in business calculations. Find the nth term formula - linear sequences You are given a sequence, and you need to find the nth term formula for each one. Lesson concludes with a pattern activity to tie everything together. It is obtained by substituting the formula for the general term into the above formula and simplifying. In other words, the difference between the adjacent terms in the arithmetic sequence is the same. Finding the nth term of an arithmetic sequence worksheet : Here we are going to see some practice questions on finding the nth term of an arithmetic sequence. 2 R E A L L I F E The nth term of an arithmetic sequence with. Equation for an Arithmetic Sequence Let a n be the nth term of an arithmetic sequence with fi rst term a 1 and common difference d. For example: 2, 4, 6, 8 You can add 2 to each number to get to the next number. Then take the reciprocal of the answer in AP to get the correct term in HP. How to find the nth term rule of a Quadratic or Cubic Sequence. Arial Calibri Default Design Microsoft Equation 3. The nth term of Zack’s sequence is three times the nth term of Livy’s sequence. What is , the first term of the sequence? If you said -5, give yourself a high five. An arithmetic sequence is a sequence in which every term after the first term is found by adding a constant – called the common difference (d). Arithmetic Progression: Definition 1: An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one. Basically we need to find two things, the first term of the sequence and the common difference, d. The nth partial sum of an arithmetic sequence can be calculated using the first and last terms as follows: S n = n. The Arithmetic series of finite number is the addition of numbers and the sequence that is generally followed include - (a, a + d, a + 2d, …. 5 11, 17, 23, 29, In an arithmetic sequence, the difference between consecutive terms. 6, 10, 14 - Answered by a verified Math Tutor or Teacher We use cookies to give you the best possible experience on our website. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. with the last term l and common ratio r is l/(r (n-1)). So let's write it like this in a table. If we multiply, it is a geometric sequence. Instructions: This algebra calculator will allow you to compute elements of an arithmetic sequence. An Arithmetic Sequence is a sequence whose consecutive terms are obtained by the addition of a constant value. What is the common difference for Livy’s sequence? Show or explain how you got your answer. nth term introduction and RAG task, followed by another RAG task to find the first 5 terms of a linear sequence. For instance. If you know the first few terms of an arithmetic sequence, you can write a general expression for the sequence to find the nth term. You can derive an arithmetic sequence formula that allows you to calculate the nth term in any sequence. Which of the following sequences are arithmetic? Identify the common difference. In an Arithmetic Sequence the difference between one term and the next is a constant. KS3 / GCSE worksheet based on Fernando character. Which term is x. This arithmetic sequence calculator (also called the arithmetic series calculator) is a handy tool for analyzing a sequence of numbers that is created by adding a constant value each time. Solutions of Chapter 9 Sequences and Series of Class 11 NCERT book available free. There are infinite sequences whose domain is the set of all positive integers, and there are finite sequences whose domain is the set of the first n positive integers. Use arithmetic sequences and series in real-life problems, such as finding the number of cells in a honeycomb in Ex. Learn Java by examples. Find the sum of the first 100 odd numbers. Next term bingo starter. I will be working hard over the next couple of weeks to upload relevant resources and activate these links. Plug these values in the formula, we get. So if we have the term, just so we have things straight, and then we have the value, and then we have the value of the term. Section 1 Arithmetic Progression An arithmetic progression is a list of numbers where the di erence between successive numbers is constant. The formula provides an algebraic rule for determining the terms of the sequence. This is much easier than writing out the sequence and counting the terms by hand, especially when the sequence is long. Write an expression that can be used to find the nth term of Livy’s sequence. This formula tell us that if we know the first term a1 and the common difference d of an arithmetic progression, then we can arrive at any term we want. Terms can be separated by comma(,) , colon(:), semicolon(;) or blank space. a 38 = Given: A term in an arithmetic sequence and the common difference. Find an equation for the nth term of the arithmetic sequence. This Bingo game gives students practice of generating the first 4 terms of an arithmetic sequence from its nth term or vice versa. The nth Term of Arithmetic Sequences Bingo (1 member review) This Bingo game gives students practice of generating the first 4 terms of an arithmetic sequence from its nth term or vice versa. An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a constant. For example: 2, 4, 6, 8 You can add 2 to each number to get to the next number. y-intercept of the line. So we must subtract 2. 1 - Enter the first term A1 in the sequence, the common difference d and n the number of terms in the sum then press enter. 1 1 2 3 5 8 The rule to continue a Fibonacci sequence is, the next term in the sequence is the sum of the two previous terms. Make your hectic wok easy through installing the ideas of finding the nth term of an arithmetic sequence. x 2 = a + d. Arithmetic Progressions. Posts about sequences written by corbettmaths. Find the common difference between the terms. Since these are square numbers then the nth term of this sequence is n^2. The value of the \(n^{th}\) term of the arithmetic sequence, \(a_n\) is computed by using the following formula: \[a_n = a_1 + (n-1)d\] This means that in order to get the next element in the sequence we add \(d\), to the previous one. Many sequences have patterns. NAME:_____ Use the formulas provided to you to complete the following. Find the first term and the common difference of the arithmetic sequence described. The general formula for an arithmetic sequence is very valuable because it allows us to find the value of any term in the sequence with little work and with the use of simple mathematical concepts. a=13, d=-6, n=21 15. Arithmetic Series— 2. In your sequence, a = 5, and d = -3. Find the fifteenth term. The constant quantity stated in the above definition is called the common difference of the progression. Formula for the nth Partial Sum of an Arithmetic Sequence 3. For arithmetic sequences, we have to add d at every step: x 1 = a. This is a comprehensive guide to the Arithmetic and Geometric Series. Learning progresses from plotting and reading coordinates in the first quadrant to describing geometric sequences using the nth term. is an arithmetic sequence because 3 is being added each time to get the next term. The first term is a 1, the common difference is d, and the number of terms is n. - [Instructor] We are asked what is the value of the 100th term in this sequence, and the first term is 15, then nine, then three, then negative three. Each term should be the n th digit of \(p\). Questions cover n+x, n-x, x+n and n-x Each question consists of… A slide with the. Find the n th term of the arithmetic sequence 13,16,19… Solution: a=13 , d = 16-13 = 3. Arithmetic sequences and series. Whenever you see the nth term equal to some expression involving an (n – 1)th term, that’s a recursively defined sequence. In treating sequences, it is customary to use subscript notation instead of functional. 3 A sequence of terms u 1, u 2, u 3, …is defined by u n = 24 – n. The ﬁ rst term is 14 and the common difference is − 3. Geometric Sequences: A Formula for the’ n – th ‘ Term. Geometric Sequence: is a sequence with a common ratio, an = a1rn - 1. Find PowerPoint Presentations and Slides using the power of XPowerPoint. Finite sequence: It is defined as the sequence which has a last term. Examples :. com, find free presentations research about Finding The Nth Term Of A Sequence PPT. The first term is 20, the second is 40, and the nth term is given by the formula x = 20 times n. You will get a tremendous impact on your busy schedule as this simple idea will save plenty of time of your overall working hours. This is a comprehensive guide to the Arithmetic and Geometric Series. Arithmetic Progressions. In other words,. a represents first term and d is common difference. Arithmetic Sequences. and so on) where a is the first term, d is the common difference between terms. x 3 = a + d + d. Similarly, in a geometric sequence, the nth term is given by 1an=a1•rn−1 , where r is the common ratio. Find the formula for the nth term. org, where students, teachers and math enthusiasts can ask and answer any math question. NAME:_____ Use the formulas provided to you to complete the following. This constant is called the common difference (d). A Sequence is a set of things (usually numbers) that are in order. An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers, whose consecutive numbers differ by a constant. In other words, quantities are said to be in arithmetic sequence when they increase or decrease by a common difference. Arithmetic Sequences – Nth Term 01. com A collection of really good online calculators for use in every day domestic and commercial use!. For instance, the sequence 5, 7, 9, 11, 13, 15,. For this sequence, the first term a_1 = 1 and the number 2008 appears somewhere in the sequence. Find its first term. The nth term is referring to any term in the arithmetic sequence. The tenth term of an arithmetic progression is equal to twice the fourth term. The denominators of each fraction are 2,3,4,5, and this is a linear sequence with nth term n + 1. S n { S }_{ n } S n is the sum of the first n n n terms of the sequence. The nth term is given by a n = a 1r n − 1. Also, it can identify if the sequence is arithmetic or geometric. The common difference is the coefficient of n. KS3 / GCSE worksheet based on Fernando character. He will then double the amout daily for each day this month that you keep the garage tidy. the indicated term in the given sequence. Name _____ Arithmetic Sequences Show all your work! Regents Questions 1. Homework problems on arithmetic sequences often ask us to find the nth term of a sequence using a formula. An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a constant. You can derive an arithmetic sequence formula that allows you to calculate the nth term in any sequence. In the beginning we will learn how to write terms for an Arithmetic or Geometric Sequence when we are given either the common difference or the common ratio. Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums. You will get a tremendous impact on your busy schedule as this simple idea will save plenty of time of your overall working hours. OK, so I have to admit that this is sort of a play on words since each element in a sequence is called a term, and we’ll talk about the terms (meaning words) that are used with sequences and series, and the notation. 2 - Finding the nth term of an arithmetic sequence To find the nth (specific) term of a sequence with a common difference you can use the formula. For instance, the sequence 5, 7, 9, 11, 13, 15,. Since these are square numbers then the nth term of this sequence is n^2. Yeah, I know why I couldn't do it at first, because I was trying to be be overly clever and do it and one formula and in one cell so I could type in "N2" or "N+1" or "5N" or "N2-7" etcetera. Arithmetic Sequences - Concept. 1) -19, Infinite Algebra 2 - Arithmetic Sequences: Finding the nth Term. Each term after the first is obtained by adding a constant, d to the preceding term, and is called the common difference. It will be part of your formula much in the same way x's and y's are part of algebraic equations. Write an Equation for the nth Term Write an equation for the nth term of the arithmetic sequence 8, 17, 26, 35, …. Usually, the formula for the nth term of an arithmetic sequence whose first term is a 1 and whose common difference is d is displayed below. nth term introduction and RAG task, followed by another RAG task to find the first 5 terms of a linear sequence. If a 1 is the first term of an arithmetic sequence and d. Arithmetic Sequences: A Quick Intro; Geometric Sequences: A Formula for the’ n – th ‘ Term. You would figure out the formula an = a1+(n-1)d-10 where an is your y-value, a1 is your first term in a number sequence (your x. The missing terms in an arithmetic sequence are called "arithmetic means". The notation a 1, a 2, a 3,… a n is used to denote the different terms in a sequence. Leonardo Fibonacci, who was born in the 12th century, studied a sequence of numbers with a different type of rule for determining the next number in a sequence. The fact-checkers, whose work is more and more important for those who prefer facts over lies, police the line between fact and falsehood on a day-to-day basis, and do a great job. Today, my small contribution is to pass along a very good overview that reflects on one of Trump’s favorite overarching falsehoods. Namely: Trump describes an America in which everything was going down the tubes under Obama, which is why we needed Trump to make America great again. And he claims that this project has come to fruition, with America setting records for prosperity under his leadership and guidance. “Obama bad; Trump good” is pretty much his analysis in all areas and measurement of U.S. activity, especially economically. Even if this were true, it would reflect poorly on Trump’s character, but it has the added problem of being false, a big lie made up of many small ones. Personally, I don’t assume that all economic measurements directly reflect the leadership of whoever occupies the Oval Office, nor am I smart enough to figure out what causes what in the economy. But the idea that presidents get the credit or the blame for the economy during their tenure is a political fact of life. Trump, in his adorable, immodest mendacity, not only claims credit for everything good that happens in the economy, but tells people, literally and specifically, that they have to vote for him even if they hate him, because without his guidance, their 401(k) accounts “will go down the tubes.” That would be offensive even if it were true, but it is utterly false. The stock market has been on a 10-year run of steady gains that began in 2009, the year Barack Obama was inaugurated. But why would anyone care about that? It’s only an unarguable, stubborn fact. Still, speaking of facts, there are so many measurements and indicators of how the economy is doing, that those not committed to an honest investigation can find evidence for whatever they want to believe. Trump and his most committed followers want to believe that everything was terrible under Barack Obama and great under Trump. That’s baloney. Anyone who believes that believes something false. And a series of charts and graphs published Monday in the Washington Post and explained by Economics Correspondent Heather Long provides the data that tells the tale. The details are complicated. Click through to the link above and you’ll learn much. But the overview is pretty simply this: The U.S. economy had a major meltdown in the last year of the George W. Bush presidency. Again, I’m not smart enough to know how much of this was Bush’s “fault.” But he had been in office for six years when the trouble started. So, if it’s ever reasonable to hold a president accountable for the performance of the economy, the timeline is bad for Bush. GDP growth went negative. Job growth fell sharply and then went negative. Median household income shrank. The Dow Jones Industrial Average dropped by more than 5,000 points! U.S. manufacturing output plunged, as did average home values, as did average hourly wages, as did measures of consumer confidence and most other indicators of economic health. (Backup for that is contained in the Post piece I linked to above.) Barack Obama inherited that mess of falling numbers, which continued during his first year in office, 2009, as he put in place policies designed to turn it around. By 2010, Obama’s second year, pretty much all of the negative numbers had turned positive. By the time Obama was up for reelection in 2012, all of them were headed in the right direction, which is certainly among the reasons voters gave him a second term by a solid (not landslide) margin. Basically, all of those good numbers continued throughout the second Obama term. The U.S. GDP, probably the single best measure of how the economy is doing, grew by 2.9 percent in 2015, which was Obama’s seventh year in office and was the best GDP growth number since before the crash of the late Bush years. GDP growth slowed to 1.6 percent in 2016, which may have been among the indicators that supported Trump’s campaign-year argument that everything was going to hell and only he could fix it. During the first year of Trump, GDP growth grew to 2.4 percent, which is decent but not great and anyway, a reasonable person would acknowledge that — to the degree that economic performance is to the credit or blame of the president — the performance in the first year of a new president is a mixture of the old and new policies. In Trump’s second year, 2018, the GDP grew 2.9 percent, equaling Obama’s best year, and so far in 2019, the growth rate has fallen to 2.1 percent, a mediocre number and a decline for which Trump presumably accepts no responsibility and blames either Nancy Pelosi, Ilhan Omar or, if he can swing it, Barack Obama. I suppose it’s natural for a president to want to take credit for everything good that happens on his (or someday her) watch, but not the blame for anything bad. Trump is more blatant about this than most. If we judge by his bad but remarkably steady approval ratings (today, according to the average maintained by 538.com, it’s 41.9 approval/ 53.7 disapproval) the pretty-good economy is not winning him new supporters, nor is his constant exaggeration of his accomplishments costing him many old ones). I already offered it above, but the full Washington Post workup of these numbers, and commentary/explanation by economics correspondent Heather Long, are here. On a related matter, if you care about what used to be called fiscal conservatism, which is the belief that federal debt and deficit matter, here’s a New York Times analysis, based on Congressional Budget Office data, suggesting that the annual budget deficit (that’s the amount the government borrows every year reflecting that amount by which federal spending exceeds revenues) which fell steadily during the Obama years, from a peak of $1.4 trillion at the beginning of the Obama administration, to $585 billion in 2016 (Obama’s last year in office), will be back up to $960 billion this fiscal year, and back over $1 trillion in 2020. (Here’s the New York Times piece detailing those numbers.) Trump is currently floating various tax cuts for the rich and the poor that will presumably worsen those projections, if passed. As the Times piece reported: