# Lesson BASICS - Arithmetic Series

Algebra ->  Algebra  -> Sequences-and-series -> Lesson BASICS - Arithmetic Series      Log On

 Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help! Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!

 Algebra: Sequences of numbers, series and how to sum them Solvers Lessons Answers archive Quiz In Depth
 This Lesson (BASICS - Arithmetic Series) was created by by longjonsilver(2297)  : View Source, ShowAbout longjonsilver: I have a new job in September, teaching Introduction Arithmetic Series or Progressions are sequences of numbers that increment by a fixed common difference eg 4,7,10,13,16,... is an Arithmetic series with common difference 3. Definitions Let d = common difference Let a = first term Let n = number of terms Let Sn = sum of first n terms So, in the sequence -2,3,8,13,18,... we have: first term is -2 --> a = -2 difference is 5 --> d = 5 nth Term Algebraically: 1st term = a 2nd term = a + d 3rd term = a + 2d 4th term = a + 3d 5th term = a + 4d 6th term = a + 5d and so on. In general, the nth term is given as a + (n-1)d --> EXAMPLES Q. Find the 12th term of the sequence 1,5,9,13,... A. a=1, d=4, n=12. which we can verify by writing out the first 12 terms in full: 1,5,9,13,17,21,25,29,33,37,41,45 Summation The summation of n terms is given by: which is also written as where L is the last term, defined as L=a+(n-1)d EXAMPLES Q. Find the sum of the first 10 terms of the sequence 1,5,9,13,17,... a=1, d=4, n=10 --> Sn = 190 Further Example Q. The second term of an arithmetic series is 5 and the fifth term is 14. Find the common difference, the tenth term and the sum of the first 8 terms. We do not know a, which is crucial in all the calculations, so we are aiming to find that as well as d. So, what we do know is the following: 2nd term, a+d = 5 5th term, a+4d = 14 subtract these to give 3d = 9 --> d = 3 So, using this in a+d=5 we have a+3 = 5 --> a = 2 The 10th term = a+9d The 10th term = 2+9(3) The 10th term = 2+27 The 10th term = 29 Now for the sum of 8 terms: Sn = 100 Summary This is the introduction to Arithmetic Series'. However, with the question quoted at the end here, there is not a lot of things that you need to know. It is all about practice now: practice using the "nth term" and "summation" formulae. This lesson has been accessed 13477 times.