Arithmetic Progression

1. Arithmetic progresssion: Arithmetic progression(AP) or arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, d to the preceding term. The constant d is called common difference.

An arithmetic progression is given by:

a, (a + d), (a + 2d), (a + 3d), ...

where a = the first term , d = the common difference

n^th term of an arithmetic progression

t_n = a + (n – 1)d

where t_n = n^th term, a = the first term, d = common difference

Number of terms of an arithmetic progression:

n=(l−a)d+1">n=(l−a)/d+1

where n = number of terms, a= the first term , l = last term, d= common differenc

Sum of first n terms in an arithmetic progression:

Sn=n2[ 2a+(n−1)d ] =n2(a+l)">Sn=n/2[ 2a+(n−1)d ] =n/2(a+l)

where a = the first term,
d= common difference,
l">l = t_n = n^th term = a + (n-1)d

2. Arithmetic Mean: If a, b, c are in AP, b is the Arithmetic Mean (AM) between a and c. In this case, b=12(a+c)">b=1/2(a+c)

b=12(a+c)">The Arithmetic Mean (AM) between two numbers a and b = 12(a+b)">1/2(a+b)

Solve most of the problems related to AP, the terms can be conveniently taken as:

3 terms: (a – d), a, (a +d)
4 terms: (a – 3d), (a – d), (a + d), (a +3d)
5 terms: (a – 2d), (a – d), a, (a + d), (a +2d)

T_n = S_n - S_n-1

If each term of an AP is increased, decreased, multiplied or divided by the same non-zero constant, the resulting sequence also will be in AP.

In an AP, the sum of terms equidistant from beginning and end will be constant.