Difference between revisions of "Arithmetic series"

Revision as of 18:50, 17 February 2016

An arithmetic series is a sum of consecutive terms in an arithmetic sequence. For instance,

$2 + 6 + 10 + 14 + 18$

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 ( $S$ is the sum, $a$ is the first term, $z$ is the last term, and $d$ is the common difference): $S = a + (a+d) + (a+2d) + \ldots + (z-d) + z$ Flipping the right side of the equation we get $S = z + (z-d) + (z-2d) + \ldots + (a+d) + a$

Now, adding the above two equations vertically, we get

$2S = (a+z) + (a+z) + (a+z) + ... +(a+z) + (a+z)$

This equals $2S = n(a+z)$ , so the sum is $\frac{n(a+z)}{2}$ , where $n$ is the number of terms.

@@ Line 4: / Line 4: @@
 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):
+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
@@ Line 16: / Line 16: @@
 Now, adding the above two equations vertically, we get
-<cmath>2S = (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 ==

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Difference between revisions of "Arithmetic series"

Revision as of 18:50, 17 February 2016

Contents

Formula

Problems

Introductory Problems

Intermediate Problems

Olympiad Problem

See also