PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: High Entry average rating: No information on entry rating
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:

$\textstyle \parbox{\linewidth}{\begin{align*} S&=\makebox[7em]{$(a_1+0)$}+\make... ...b+\makebox[7em]{$(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 equals to $ (2a_1+(n-1)d)$. The sum is then

$\displaystyle S=\frac{(2a_1+(n-1)d)n}{2}.$    



"arithmetic progression" is owned by bbukh.
(view preamble)

View style:

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


Attachments:
sum of odd numbers (Example) by pahio
Log in to rate this entry.
(view current ratings)

Cross-references: Gauss', sum, sequence, difference, term, length
There are 6 references to this entry.

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 16764 times total.

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

Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)