Progression

When in a sequence, a term is related to another term in mathematical way, then this sequence is called Progression. Progression is of 3 kinds: Arithmetic progression, Geometric progression, Harmonic progression.

Arithmetic Progression:

An arithmetic progression is a sequence of numbers where the difference of any two successive members is a constant. This constant could be positive or negative numbers. This constant is considered as 'common deference' of that sequence. Common difference is denoted by 'd'.

For instance, the sequence 1, 3, 5, 7, 9, 11, 13 … is an arithmetic progression with common difference of 2.

Similarly  50, 40, 30, 20, 10, 0, -10, -20.....  is an arithmetic progression with common difference -10.

To make sure whether a sequence or series is arithmetic or not, we just need to subtract each term from its successive term(except the first term). If the result of subtraction is same in all cases, then it is confirmed that the sequence or series is arithmetic.

If the initial term of an arithmetic progression is a₁ and the common difference of successive members is d, then the nth term of the sequence (a_n) is given by:

So, it advances in following way:

a, a+d, a+2d,  a+3d, a+4d. ..........

If d is the common difference and  we denote the n-th term of an arithmetic progression as a_n.  By S_n we denote the sum of the first n elements of an arithmetic series.

Arithmetic series means the sum of the elements of an arithmetic progression.

Then summation S_n can be obtained by :

Geometric Progression:

A geometric progression, is a sequence of numbers where each term (except the first one) is found by multiplying the previous one by a fixed number called the common ratio.

For instance, the sequence 3, 6, 12, 24, ... is a geometric progression with common ratio 2.

Similarly 1, 1/2, 1/4, 1/8 ... is a geometric sequence with common ratio 1/2.

In General we write a Geometric Sequence like this:

a, ar, ar², ar³, ...

where a is the first term and r ≠ 0 is the common ratio.

The n-th term of a geometric progression can be found by:   

a_n = ar^{n-1} 

where a is initial value and common ratio  is r.

A geometric series is the sum of the numbers in a geometric progression and it can be found by:

 

Harmonic Progression: 

Harmonic progression is obtained from arithmetic progression.  If a sequence is in arithmetic progression, then the sequence obtained by taking the reciprocal of every term in the sequence makes Harmonic Progression.

For example, if arithmetic progression is 3, 6, 9, 12 ..... then harmonic progression is

1/3, 1/6, 1/9, 1/12 ..........

Problems related to harmonic progression are generally solved by converting it into an arithmetic progression.

Relationship between Arithmetic, Harmonic and Geometric Progression:
Let a,b be two positive real numbers.
Then AM x HM=GM².