Progressions in Maths

What are Progressions in Maths?

Progression is a list of numbers that follow a particular pattern, that can be represented via a formula.

Types of Progressions

There are three kinds of progressions that you will encounter in Maths. These are:

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

Let’s study them in more detail.

Arithmetic Progression

Arithmetic Progression is a sequence of numbers, such that there is a common difference between any two adjacent terms.

Let the first term of an arithmetic progression be ‘a’, and the common difference be ‘d’. So, the arithmetic progression will be:
a, a + d, a + 2d, a + 3d, …..

In other words, in an arithmetic progression each successive term is the sum of its previous term and a certain fixed number (called the common difference).

For example, the following is an arithmetic progression with the first term 2, and a common difference of 3.
2, 5, 8, 11, 14, 17 …..

Arithmetic Progression Formulae

Common difference

If we have an arithmetic progression: $a_1, a_2, a_3, ….. a_{n - 1}, a_n$

Then, Common difference of the Arithmetic Progression,
d = $a_2 – a_1 = a_3 - a_2 = ….. = a_n - a_{n - 1}$

• If the common difference is positive, then each subsequent term of the arithmetic series will be more than the previous one.
• If the common difference is negative, then each subsequent term of the arithmetic series will be less than the previous one.

nth Term of Arithmetic Progression

Let the first term of an arithmetic progression be ‘a’, the nth term be ‘$a_n$’, and the common difference be ‘d’.

Then the nth term, $a_n$ = a + (n - 1) d

Q. Find the 10th term of AP: 1, 2, 3, 4, 5 ……

Explanation:

Sum of the terms of an Arithmetic Progression

Formula 1

If the first term of an arithmetic progression is ‘a’, the common difference is ‘d’ and the number of terms is ‘n’, then:

Sum of n terms of an arithmetic progression = $\frac{n}{2}$ [2a + (n − 1) × d]

Formula 2

If the first term of an arithmetic progression is ‘a’, and the last term is ‘l’, then:

Sum of n terms of an arithmetic progression = $\frac{n}{2}$ (first term + last term) = $\frac{n}{2}$ (a + l)

Q. Find the sum of the elements of the following series: 8, 9, 10, 11, 12, 13, 14

Explanations :
1 2

Arithmetic Mean

Arithmetic Mean = Sum of all terms in the AP / Number of terms in the AP

Geometric Progression

Geometric Progression is a sequence of numbers, such that there is a common ratio between any two adjacent terms.

Let the first term of a geometric progression be ‘a’, and the common ratio be ‘r’. So, the geometric progression will be:
a, ar, a$r^2$, a$r^3$, …..

In other words, in a geometric progression each successive term is the multiplication of its previous term and a certain fixed number (called the common ratio).

For example, the following is a geometric progression with the first term 2, and a common ratio of 3.
2, 6, 18, 54, 162, …..

Geometric Progression Formulae

nth Term of Geometric Progression

Let the first term of a geometric progression be ‘a’, the nth term be ‘$a_n$’ and the common ratio be ‘r’.

Then the nth term, $a_n = a r^{n - 1}$

Sum of the terms of a Geometric Progression

Formula 1

If the first term of a geometric progression is ‘a’, the common ratio is ‘r’ and the number of terms is ‘n’, then:

Sum of n terms of a geometric progression (if r > 1) = $\frac{a (r^n – 1)}{r – 1}$

Sum of n terms of a geometric progression (if r < 1) = $\frac{a (1 – r^n)}{1 – r}$

Formula 2

If a geometric progression has infinite terms, and r > 1, then each subsequent term in the series will be greater in value than the previous one. So, the sum of the terms of such a series would be infinite too.

However, if a geometric progression has infinite terms, and −1 < r < 1, then the sum of the terms of such a series would be finite. We can find it using the following formula.

Sum of infinite terms of a geometric progression (if −1 < r < 1) = $\frac{a}{1 – r}$

Q. Find the sum of the geometric series 64, 32, 16, 8, 4, ….. upto infinity.

Explanation:

Geometric Mean

Geometric Mean = nth root of the product of n terms in the Geometric Progression

Harmonic Progression

Harmonic Progression is obtained by taking the reciprocal of the terms of an arithmetic progression. For example, the following is an arithmetic progression:

2, 5, 8, 11, 14, 17 ….. (with the first term 2, and a common difference of 3)

So, the harmonic progression will be:
1/2, 1/5, 1/8, 1/11, 1/14, 1/17, …..

Harmonic Progression Formulae

nth Term of Harmonic Progression

Let the first term of a harmonic progression be ‘a’, the nth term be ‘$a_n$’ and the common difference be ‘d’.

Then the nth term, $a_n = \frac{1}{[a + (n - 1) d]}$

Harmonic Mean

For two terms ‘a’ and ‘b’, Harmonic Mean = $\frac{2 ab}{a + b}$