What is Progression? Types, Examples, Formulae

The series and sequences of mathematics algebra that are connected to numbers and algebraic operations are referred to as progression.  

What is Progression?

A progression is a set of numbers (or things) that follow a specific pattern. A progression is sometimes referred to as a sequence. Every phrase in a progression is generated by applying a certain rule to the number before it. In other words, a general term (or) nth term, represented by an. 

For example, the natural number sequence 1, 2, 4, 6, 8… is an Arithmetic Progression with a common difference of two between two subsequent terms.  

Types of Progressions

Arithmetic progression (AP) 

Geometric progression (GP) 

Harmonic progression (HP) 

What is Arithmetic Progression?

An arithmetic progression (AP) is a numerical series in which each consecutive term is the sum of the term before it and a fixed integer. The common difference is the name given to this fixed number. For instance, 1, 4, 7, 10… is an AP because each number is produced by adding a fixed number 3 to its preceding phrase.

2nd semester = 4 = 1 + 3 = 1st semester + 3 

3rd semester = 7 = 4 + 3 = 2nd semester + 3 

Fourth term = 10 = 7 + 3 = third term + 3, and so on. 

Arithmetic Progression Examples

Example 1: 

Determine the value of n if a = 10, d = 5, and an = 95. 

Given, a = 10, d = 5, and an = 95. 

Using formula: 

a = a + (n − 1) × d

95 = 10 + (n − 1) × 5 

(n − 1) × 5 = 95 – 10 = 85 

(n − 1) = 85/ 5 

(n − 1) = 17 

n = 17 + 1 

n = 18 

Example 2: 

Find the 10th term of the AP 2,12, 22, 32…. 

Solution: 

Here first term a = 2, and the common difference d is 12-2 = 10 = d. 

Using the formula and putting n=10, we get 

a10 = 2+ (10-1)10 = 2 + 90 = 92. 

Arithmetic progression Formulae

Let a be the first term of the progression, d be a common difference, and a be the nth term. The arithmetic progression formulae are thus as follows: a = a + (n – 1) d 

d = a – an-1 

\(S_{n} = \frac{n}{2} (2a+(n-1)d)\)

or

\(S_{n}= \frac{n}{2}(a+I)\)

where l is the last term and equals Tn

First Term of Arithmetic Progression

The Arithmetic Progression can alternatively be stated in terms of common differences, as follows: a, a + d, a + 2d, a + 3d, a + 4d, ……….., a + (n – 1) d, where “a” is the progression’s first term. 

What is Geometric Progression?

A geometric progression (GP) is a numerical series in which each consecutive term is the product of the term before it and a fixed integer. The common ratio is the name given to this constant quantity. For instance, 4, 16, 64, 256… is a GP because each number is produced by multiplying a fixed integer 4 by its preceding term.  

2nd term = 16 = 4(4) = 4 (1st term) 

The third term = 64 = 4(16) = 4 (2nd term) 

4th term = 256 = 4(64) = 4(3rd term), etc. 

Geometric Progression Formulae

Let a be the first term of the progression, r be the common ratio, and a be the nth term. The geometric progression formulae are then provided by:  

\(a_{n}=a. r^{n-1}\)

\(S_{n}= a\frac{( r^{n}-1 )}{r-1}\) (when r is 1 and Sn = na when r is 1.)

When |r| is 1, the sum of infinite geometric series,

\(S= \frac{a}{1-r}\)

and, S diverges when |r| is 1.

Geometric Progression Example

Consider the following geometric progression: 1, 4, 16, 64… Keep in mind that 4/1 = 16/4 = 64/16 =… = 4. All of the ratios are the same. As a result, it is a GP.  

What is Harmonic Progression?

A harmonic progression is a series formed by taking the reciprocal of an arithmetic progression’s terms. A natural number of series is an arithmetic progression. Therefore, we obtain 1,1/2,1/3,1/4… by calculating the reciprocals of each term. This is an example of harmonic progression.  

Harmonic Progression Formula

In the case of a harmonic progression 1/a, 1/(a+d), 1/(a+2d) … 

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

Sum of the first terms,  \(S_{n}= \frac{1}{d} ln [\frac{2a + (2n – 1) d}{2a-d}]\)

Harmonic Progression Example

In HP, 

1/2,1/4, 1/6, 1/8, 1/16 

You can see that if you write the denominators individually, they fit in the AP style. 

Final Notes

As a result, harmonic progressions terms are those whose denominators are in the arithmetic progression in the correct order and have the same common difference.  

We addressed the equations, progression examples, definitions of progressions, types of progressions, and the distinction between Arthematic Progression, Geometric Progression, and Harmonic Progression. If you have any doubts about progressions or other issues, as well as in sorting out formulae. Then do look at Tutoroot’s Maths online Tuition. The skilled professors will give access to the ideal atmosphere, allowing you to understand all the complicated topics.