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== | ||
+ | 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> | ||
+ | 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>, where <math>n</math> is the number of terms. | |
− | |||
− | This equals <math>2S = n( | ||
== Problems == | == Problems == |
Revision as of 18:50, 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
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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