arithmetic progression

Arithmetic progressionMathworldPlanetmathPlanetmath of length n, initial term a1 and common difference d is the sequenceMathworldPlanetmath a1,a1+d,a1+2d,,a1+(n-1)d.

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

S =(a1+0)+(a1+d)++(a1+(n-2)d)+(a1+(n-1)d) +S¯ =(a1+(n-1)d)+(a1+(n-2)d)++(a1+d)+(a1+0)¯ 2S =(2a1+(n-1)d)+(2a1+(n-1)d)++(2a1+(n-1)d)+(2a1+(n-1)d).

We just add the sum with itself written backwards, and the sum of each of the columns equals to (2a1+(n-1)d). The sum is then

Title arithmetic progression
Canonical name ArithmeticProgression
Date of creation 2013-03-22 13:39:00
Last modified on 2013-03-22 13:39:00
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 10
Author bbukh (348)
Entry type Definition
Classification msc 00A05
Classification msc 11B25
Related topic MulidimensionalArithmeticProgression
Related topic SumOfKthPowersOfTheFirstNPositiveIntegers