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'arithmetic progression'
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| Title of object: |
arithmetic progression |
| Canonical Name: |
ArithmeticProgression |
| Type: |
Definition |
| Created on: |
2003-05-26 15:20:40 |
| Modified on: |
2003-10-22 01:12:16 |
| Classification: |
msc:00A05, msc:11B25 |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic} |
Content:
Arithmetic progression of length $n$, initial term $a_1$ and common difference $d$ is the sequence $a_1, a_1+d,a_1+2d,\dotsc,a_1+(n-1)d$.
%%%This code is not displayed by PlanetMath... seems to be a bug in LaTeX2HTML
%%% Added later: parbox fixes it since it forces LaTeX2HTML to render the whole
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The sum of terms of an arithmetic progression can be computed using Gauss's trick:
\parbox{\linewidth}{\begin{align*}
S&=\makebox[7em]{$(a_1+0)$}+\makebox[7em]{$(a_1+d)$}+\dotsb+\makebox[7em]{$(a_1+(n-2)d)$}
+\makebox[7em]{$(a_1+(n-1)d)$}\\
+\underline{S\vphantom{\makebox[7em]{$(a_1+(n-1)d)$}}}&\underline{{}=\makebox[7em]{$(a_1+(n-1)d)$}+
\makebox[7em]{$(a_1+(n-2)d)$}+\dotsb+\makebox[7em]{$(a_1+d)$}
+\makebox[7em]{$(a_1+0)$}}\\
2S&=\makebox[7em]{$(2a_1+(n-1)d)$}+\makebox[7em]{$(2a_1+(n-1)d)$}+\dotsb+\makebox[7em]{$(2a_1+(n-1)d)$}+
\makebox[7em]{$(2a_1+(n-1)d)$}.
\end{align*}}
We just add the sum with itself written backwards, and the sum of each of the columns equal 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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