|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| − | An '''arithmetic series''' is a sum of consecutive terms in an [[arithmetic sequence]]. For instance,
| + | #REDIRECT[[Arithmetic sequence]] |
| − | | |
| − | <math> 2 + 6 + 10 + 14 + 18 </math>
| |
| − | | |
| − | is an arithmetic series whose value is 50.
| |
| − | ==Formula==
| |
| − | To find the sum of an arithmetic sequence, we can write it out in two as so (<math>S</math> is the sum, <math>a</math> is the first term, <math>z</math> is the number of terms, and <math>d</math> is the common difference):
| |
| − | <cmath>
| |
| − | S = a + (a+d) + (a+2d) + \ldots + (z-d) + z
| |
| − | </cmath>
| |
| − | Flipping the right side of the equation we get
| |
| − | <cmath>
| |
| − | S = z + (z-d) + (z-2d) + \ldots + (a+d) + a
| |
| − | </cmath>
| |
| − | | |
| − | Now, adding the above two equations vertically, we get
| |
| − | | |
| − | <cmath>2S = (a+z) + (a+z) + (a+z) + ... +(a+z) + (a+z)</cmath>
| |
| − | | |
| − | This equals <math>2S = n(a+z)</math>, so the sum is <math>\frac{n(a+z)}{2}</math>.
| |
| − | | |
| − | == Problems ==
| |
| − | === Introductory Problems ===
| |
| − | * [[2006_AMC_10A_Problems/Problem_9 | 2006 AMC 10A, Problem 9]]
| |
| − | *[[2006 AMC 12A Problems/Problem 12 | 2006 AMC 12A, Problem 12]]
| |
| − | | |
| − | === Intermediate Problems ===
| |
| − | *[[2003 AIME I Problems/Problem 2|2003 AIME I, Problem 2]]
| |
| − | | |
| − | === Olympiad Problem ===
| |
| − | | |
| − | == See also ==
| |
| − | * [[Series]]
| |
| − | * [[Summation]]
| |
| − | | |
| − | {{stub}}
| |
