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. | ||
==Formula== | ==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 | + | 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 last term, and <math>d</math> is the common difference): |
<cmath> | <cmath> | ||
S = a + (a+d) + (a+2d) + \ldots + (z-d) + z | S = a + (a+d) + (a+2d) + \ldots + (z-d) + z | ||
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<cmath>2S = (a+z) + (a+z) + (a+z) + ... +(a+z) + (a+z)</cmath> | <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>. | + | This equals <math>2S = n(a+z)</math>, so the sum is <math>\frac{n(a+z)}{2}</math>, where <math>n</math> is the number of terms. |
== Problems == | == Problems == |
Revision as of 17:50, 17 February 2016
An arithmetic series is a sum of consecutive terms in an arithmetic sequence. For instance,
is an arithmetic series whose value is 50.
Contents
[hide]Formula
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 last term, and is the common difference): Flipping the right side of the equation we get
Now, adding the above two equations vertically, we get
This equals , so the sum is , where is the number of terms.
Problems
Introductory Problems
Intermediate Problems
Olympiad Problem
See also
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