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arithmetic progression (Definition)

Arithmetic progression of length $n$ , initial term $a_1$ and common difference $d$ is the sequence $ a_1, a_1+d,a_1+2d,\dotsc,a_1+(n-1)d$ .

The sum of terms of an arithmetic progression can be computed using Gauss's trick:

$\displaystyle S$ $\displaystyle =\makebox[7em]{$(a_1+0)$}+\makebox[7em]{$(a_1+d)$}+\dotsb+\makebox[7em]{$(a_1+(n-2)d)$} +\makebox[7em]{$(a_1+(n-1)d)$}$    
$\displaystyle +\underline{S\vphantom{\makebox[7em]{$(a_1+(n-1)d)$}}}$ $\displaystyle \underline{{}=\makebox[7em]{$(a_1+(n-1)d)$}+ \makebox[7em]{$(a_1+(n-2)d)$}+\dotsb+\makebox[7em]{$(a_1+d)$} +\makebox[7em]{$(a_1+0)$}}$    
$\displaystyle 2S$ $\displaystyle =\makebox[7em]{$(2a_1+(n-1)d)$}+\makebox[7em]{$(2a_1+(n-1)d)$}+\dotsb+\makebox[7em]{$(2a_1+(n-1)d)$}+ \makebox[7em]{$(2a_1+(n-1)d)$}.$    

We just add the sum with itself written backwards, and the sum of each of the columns equals to $(2a_1+(n-1)d)$ . The sum is then \begin{equation*} S=\frac{(2a_1+(n-1)d)n}{2}. \end{equation*}



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See Also: multidimensional arithmetic progression, sum of $r$th powers of the first $n$ positive integers


Attachments:
sum of odd numbers (Example) by pahio
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Cross-references: Gauss', sum, sequence, difference, term, length
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This is version 7 of arithmetic progression, born on 2003-05-26, modified 2004-09-24.
Object id is 4302, canonical name is ArithmeticProgression.
Accessed 21121 times total.

Classification:
AMS MSC00A05 (General :: General and miscellaneous specific topics :: General mathematics)
 11B25 (Number theory :: Sequences and sets :: Arithmetic progressions)

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