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, n is the number of terms, and d is the common difference):
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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>\begin{align*}
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<cmath>
S &=  a + (a+d) + (a+2d) + ... + (a+(n-1)d) \\
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S =  a + (a+d) + (a+2d) + ... + (z-d) + z
S &= (a+(n-1)d) + (a+(n-2)d)+ ... + (a+d) + a
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</cmath>
\end{align*}</cmath>
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Flipping the right side of the equation we get
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<cmath>
 +
S = + (z-d) + (z-2d) +...   + (a+d)   + a
 +
</cmath>
  
Now, adding vertically and shifted over one, we get
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Now, adding the above two equations vertically, we get
  
<cmath>2S = (2a+(n-1)d)+(2a+(n-1)d)+(2a+(n-1)d)+...+(2a+(n-1)d)</cmath>
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<cmath>2S = (a+z) + (a+z) + (a+z) + ... + (a+z)</cmath>
  
This equals <math>2S = n(2a+(n-1)d)</math>, so the sum is <math>\frac{n}{2} (2a+(n-1)d)</math>.
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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,

$2 + 6 + 10 + 14 + 18$

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

Now, adding the above two equations vertically, we get

\[2S = (a+z) + (a+z) + (a+z) + ... + (a+z)\]

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

Problems

Introductory Problems

Intermediate Problems

Olympiad Problem

See also

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