Difference between revisions of "Arithmetic series"
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is an arithmetic series whose value is 50. | is an arithmetic series whose value is 50. | ||
| − | To find the sum of an arithmetic sequence, we can write it out as so (S is the sum, a is the first term, | + | 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> | + | <cmath> |
| − | S | + | S = a + (a+d) + (a+2d) + ... + (z-d) + z |
| − | S | + | </cmath> |
| − | + | Flipping the right side of the equation we get | |
| + | <cmath> | ||
| + | S = z + (z-d) + (z-2d) +... + (a+d) + a | ||
| + | </cmath> | ||
| − | Now, adding vertically | + | Now, adding the above two equations vertically, we get |
| − | <cmath>2S = ( | + | <cmath>2S = (a+z) + (a+z) + (a+z) + ... + (a+z)</cmath> |
| − | This equals <math>2S = n( | + | This equals <math>2S = n(a+z)</math>, so the sum is <math>\frac{n(a+z)}{2}</math>. |
== Problems == | == Problems == | ||
Revision as of 19:26, 19 September 2015
An arithmetic series is a sum of consecutive terms in an arithmetic sequence. For instance,
is an arithmetic series whose value is 50.
To find the sum of an arithmetic sequence, we can write it out in two as so (
is the sum, 
is the first term, 
is the number of terms, and 
is the common difference):
![\[S = a + (a+d) + (a+2d) + ... + (z-d) + z\]](http://latex.artofproblemsolving.com/5/6/b/56b66e2ce192f9d136e0f8968018271e0329eda1.png)
Flipping the right side of the equation we get
![\[S = z + (z-d) + (z-2d) +... + (a+d) + a\]](http://latex.artofproblemsolving.com/0/8/5/0858de9fa94cc856f0967e7660c7f2d3772a3d4e.png)
Now, adding the above two equations vertically, we get
![\[2S = (a+z) + (a+z) + (a+z) + ... + (a+z)\]](http://latex.artofproblemsolving.com/1/9/e/19e2e0aacc430f5a9e7a4230c09f9f05d839e773.png)
This equals 
, so the sum is 
.
Contents
Problems
Introductory Problems
Intermediate Problems
Olympiad Problem
See also
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