# Progression || A.P., G.P., H.P.

## Progression Formulas and definitions

In Mathematics progression is a list or series of numbers that shows a fixed pattern and it is also called sequence. By this order or pattern, we can find the next term of the series.

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### Types of Progression

In maths, Mainly three types of progression we use,

- Arithmetic Progression (AP)
- Geometric Progression (GP)
- Harmonic Progression (HP)

### Arithmetic Progression

In Arithmetic progression, the difference between any two consecutive terms of
the series is always the same and this difference is called the
**common difference **denoted by 'd' and the **first term** of the
series is called 'a'

**e.g.-** 1, 3, 5, 7, 9.......

3-1 = 2

5-3 = 2

7-5 = 2

9-7 = 2

So, this series in AP and the **common difference** d = 2 and the
**first term** is a = 1.

#### General form of Arithmetic progression

a, a+d, a+2d, a+3d, ............... a+(n - 1)d ..........

1^{st} term T_{1} = a

2^{nd} term T_{2} = a + d

3^{rd} term T_{3} = a + 2d

n^{th} term T_{n} = a + (n - 1)d

#### Formulas for Arithmetic progression

👉To find n^{th} term of AP,

a_{n} or T_{n} = a + (n - 1)d

a_{n} = S_{n} - S_{n-1}

n^{th} term denoted by a_{n} or T_{n}

👉If the last term is given then n^{th} term from the last,

\(a_l = l - (n - 1)d\)

👉To find the sum of all n terms of AP,

\(S_n = \frac {n}{2} [ 2a + (n - 1)d]\)

If the last term is given

\(S_n = \frac {n}{2} [ a + l ]\)

👉To find the common difference

d = a_{n} - a_{n-1}

d = S_{2} - 2S_{1}

👉If a and b are in AP then the Arithmetic mean is,

\(A = \frac {a+b}{2}\)

Arithmetic mean denoted by 'A'.

👉If a, b, c are in AP then the Arithmetic mean of a and c is,

\(b = \frac {a+c}{2}\)

**Note- **If the terms in AP are to be considered, then they are
considered in this format.

__In Odd numbers__

like 3 terms in AP then,

a - d, a, a + d

or 5 terms in AP then,

a - 2d, a - d, a, a + d, a + 2d

__In Even numbers__

consider 2 terms in AP then,

a - d, a + d

or if 4 terms in AP then,

a - 3d, a - d, a + d, a + 3d

### Geometric Progression

In this series ratio of any two consecutive terms is always the same this
ratio is called the **common ratio**. It is denoted by 'r'. By
multiple this common ratio, we can find the next number of the series.

**e.g.**- 2, 6, 18, 54 ......................

\(\frac {6}{2} = 3\)

\(\frac {18}{6} = 3\)

\(\frac {54}{18} = 3\)

So, this series in GP and the **common ratio** r = 2 and
the **first term** is a = 2.

#### General form of Geometric Progression

a, ar, ar^{2}, ar^{3}, ............... ar^{n - 1} ...........

1^{st} term a_{1} = a

2^{nd} term a_{2} = ar

3^{rd} term a_{3} = ar^{2}

n^{th} term a_{n} = ar^{n - 1}

#### Formulas for Geometric Progression

👉To find n^{th} term of GP,

a_{n} = ar^{n - 1}

n^{th} term denoted by a_{n} or T_{n}

👉If the last term is given then n^{th} term from the last,

\(a_l = l\left ( \frac {1}{r} \right )^{n-1}\)

👉To find the sum of all n terms of GP,

\(S_n = \frac {a(r^n - 1)}{r-1} , r > 1\)

\(S_n = \frac {a(1 - r^n)}{1 - r} , r < 1\)

👉 The sum of the terms of GP with infinite terms,

a, ar, ar^{2}, ar^{3}, ............... ∞

\(S_∞ = \frac {a}{1 - r} , |r| < 1\)

👉If a and b are in GP then the Geometrical mean is,

\(G = \sqrt {ab}\)

Geometrical mean denoted by 'G'.

**Note- **If the terms in GP are to be considered, then they are
considered in this format.

__In Odd numbers__

like 3 terms in GP then,

\(\frac {a}{r}\), a, ar

or 5 terms in GP then,

\(\frac {a}{r^2} , \frac {a}{r}\), a, ar, ar^{2}

__In Even numbers__

consider 2 terms in GP then,

\(\frac {a}{r}\), ar

or if 4 terms in AP then,

\(\frac {a}{r^3} , \frac {a}{r}\), ar, ar^{3}

### Harmonic Progression

If we take the reciprocal of AP terms then we got Harmonic progression for example 1, 3, 5, 7, 9....... is in AP if we take reciprocal of each term 1, 1/3, 1/5, 1/7, 1/9 ....... this series in HP.

#### General Form of Harmonic Progression

\(\frac {1}{a}, \frac {1}{a+d}, \frac {1}{a+2d}........................\frac {1}{a+(n-1)d}\)

**Note- **you can solve all questions of this progression by reciprocal AP
series Let's with an example.

**Qus-** Find the 10th term if HP is \(\frac {1}{2}, \frac {1}{4}, \frac
{1}{6}.................\)?

**Solution-** Compatible AP- 2, 4, 6 .......... (reciprocal of each
term)

a = 2, d = 2

a_{n} = a + (n - 1)d

a_{10} = 2 + (10 - 1)2

a_{10} = 2 + 9×2

a_{10} = 20

this is 10 term of AP.

For 10 term of HP \(\frac {1}{20}\).

#### Formula of Harmonic Progression

👉To find n^{th} term of HP,

a_{n} = \(\frac {1}{a+(n-1)d}\)

👉To find the sum of all n terms of HP,

\(S_n = \frac {1}{d} ln[\frac {2a + (2n - 1)d}{2a-d}]\)

👉 If a and b in HP then Harmonic mean = \(\frac {2ab}{a+b}\)

👉 If a, b and c in HP then Harmonic mean = \(\frac {3abc}{ab+bc+ca}\)

### Relation between Arithmetic mean, Geometrical mean and Harmonic mean

AM ≥ GM ≥ HM

or

AM > GM > HM

or

GM^{2} = AM × HM

Where,

AM = Arithmetic mean

GM = Geometrical mean

HM = Harmonic mean

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