# Progressions

### Sequence and Series

A sequence is an arrangement of a number in a definite order according to some rule. A sequence containing a finite number of terms is called a finite sequence. Let a1, a2, a3, ... be the sequence, then the sum expressed as a1+a2+a3+ ... is called series.

**Progression**

A sequence of numbers that progress in a definite order according to certain definite rule is called a progression. There are two common types of progression:

- AP
- GP

### Arithmetic Progression (AP)

In an AP, each term of a progression differs from its preceding term by a constant. This constant difference is called the common difference of the AP.

The general form of an AP is: a, a+d, a+2d, a+3d, ...

1. The n^{th} term of AP: T_{n} = a + (n-1)d

2. The sum of n terms of AP: S_{n }= n/2[2a+(n-1)d] = n/2[a+l]

3. If three numbers are in arithmetic progression, the middle number is called the arithmetic mean of the other two terms.

Arithmetic Mean, b = (a+c)/2

4. If three terms are in AP, then they can be taken as (a – d), a, (a + d).

### Geometric Progression (GP)

In a GP, every term bears a constant ratio with ts preceding term. This constant factor is called the common ratio.

The general form of a GP is: a, ar, ar^{2}, ar^{3}, ...

1. T_{n} = ar^{n-1 }

2. Sum of n terms: S_{n }= a(1-r^{n})/1-r

3. For a decreasing geometric progression the sum to infinite number of terms is

S_{∞} = a/(1-r)

4. If three numbers are in geometric progression, the middle number is called the geometric mean of the other two terms.

Geometric Mean, b = √(ac)

5. If three numbers are in GP, then they can be taken as a/r, a, ar.

### Harmonic Progression (HP)

If the reciprocals of the terms of a series form an arithmetic progression, then the series is called a harmonic progression.

If a, b, c are in harmonic progression, then b = 2ac/(a+c) where b is the harmonic mean.

### Sum of Natural Series

The sum of the first n natural numbers = n(n+1)/2

The sum of the square of the first n natural numbers = (n)(n+1)(2n+1)/6

The sum of the cubes of the first n natural numbers = n^{2}(n+1)^{2}/4

The sum of first n even numbers = n(n + 1)

The sum of first n odd numbers = n^{2}