S n = a n + ( a n − d ) + ( a n − 2 d ) + … + a 1Īnd adding these two equations together, the terms involving d add to zero and we obtain n factors of a 1 + a n:Ģ S n = ( a 1 + a n ) + ( a 1 + a n ) + … + ( a n + a 1 ) 2 S n = n ( a 1 + a n )ĭividing both sides by 2 leads us the formula for the nth partial sum of an arithmetic sequence The sum of the first n terms of an arithmetic sequence given by the formula: S n = n ( a 1 + a n ) 2. Therefore, we next develop a formula that can be used to calculate the sum of the first n terms, denoted S n, of any arithmetic sequence. However, consider adding the first 100 positive odd integers. S 5 = Σ n = 1 5 ( 2 n − 1 ) = + + + + = 1 + 3 + 5 + 7 + 9 = 25Īdding 5 positive odd integers, as we have done above, is managable. For example, the sum of the first 5 terms of the sequence defined by a n = 2 n − 1 follows: is the sum of the terms of an arithmetic sequence. In some cases, the first term of an arithmetic sequence may not be given.Īn arithmetic series The sum of the terms of an arithmetic sequence. Next, use the first term a 1 = − 8 and the common difference d = 3 to find an equation for the nth term of the sequence.Ī n = − 8 + ( n − 1 ) ⋅ 3 = − 8 + 3 n − 3 = − 11 + 3 n Substitute a 1 = − 8 and a 7 = 10 into the above equation and then solve for the common difference d. In this case, we are given the first and seventh term:Ī n = a 1 + ( n − 1 ) d U s e n = 7. In other words, find all arithmetic means between the 1 st and 7 th terms.īegin by finding the common difference d. In fact, any general term that is linear in n defines an arithmetic sequence.įind all terms in between a 1 = − 8 and a 7 = 10 of an arithmetic sequence. In general, given the first term a 1 of an arithmetic sequence and its common difference d, we can write the following:Ī 2 = a 1 + d a 3 = a 2 + d = ( a 1 + d ) + d = a 1 + 2 d a 4 = a 3 + d = ( a 1 + 2 d ) + d = a 1 + 3 d a 5 = a 4 + d = ( a 1 + 3 d ) + d = a 1 + 4 d ⋮įrom this we see that any arithmetic sequence can be written in terms of its first element, common difference, and index as follows:Ī n = a 1 + ( n − 1 ) d A r i t h m e t i c S e q u e n c e Here a 1 = 1 and the difference between any two successive terms is 2. For example, the sequence of positive odd integers is an arithmetic sequence, zip file containing this book to use offline, simply click here.Īn arithmetic sequence A sequence of numbers where each successive number is the sum of the previous number and some constant d., or arithmetic progression Used when referring to an arithmetic sequence., is a sequence of numbers where each successive number is the sum of the previous number and some constant d.Ī n = a n − 1 + d A r i t h m e t i c S e q u e n c eĪnd because a n − a n − 1 = d, the constant d is called the common difference The constant d that is obtained from subtracting any two successive terms of an arithmetic sequence a n − a n − 1 = d. You can browse or download additional books there. More information is available on this project's attribution page.įor more information on the source of this book, or why it is available for free, please see the project's home page. Additionally, per the publisher's request, their name has been removed in some passages. However, the publisher has asked for the customary Creative Commons attribution to the original publisher, authors, title, and book URI to be removed. ![]() Normally, the author and publisher would be credited here. ![]() This content was accessible as of December 29, 2012, and it was downloaded then by Andy Schmitz in an effort to preserve the availability of this book. ![]() See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and do make it available to everyone else under the same terms. This book is licensed under a Creative Commons by-nc-sa 3.0 license.
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