Progression

A progression (which is also known as a sequence) is nothing but a pattern of numbers. For example, 3, 6, 9, 12, ... is a progression because there is a pattern observed where every number here is obtained by adding 3 to its previous number. But this pattern doesn't need to be the same in every progression.

The pattern of a progression depends on its type. Let us see the types of progressions along with examples and their formulas.

Progression is a list of numbers (or items) that exhibit a particular pattern. A progression is also known as a sequence. In a progression, every term is obtained by applying a specific rule on its previous term. In other words, every term of a progression is defined by a general term (or) n^th term, which is denoted by a_n.

For example, the n^th term of a progression 3, 5, 7, 9, ... is a_n = 2n + 1. Substituting n = 1, 2, 3, ... here, we get the 1^st, 2^nd, 3^rd, .... terms. For example:

• When n = 1, first term = 2(1) + 1 = 3.
• When n = 2, second term = 2(2) + 1 = 5.
• When n = 3, third term = 2(3) + 1 = 7
  and so on.

There are mainly 3 types of progressions in math. They are:

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

Each type of progression along with a simple definition and example is tabulated below.

We will learn about each progression in detail in the upcoming sections.

An arithmetic progression (AP) is a sequence of numbers in which each successive term is the sum of its preceding term and a fixed number. This fixed number is called the common difference. For example, 1, 4, 7, 10, ... is an AP as every number is obtained by adding a fixed number 3 to its previous term.

• 2^nd term = 4 = 1 + 3 = 1^st term + 3
• 3^rd term = 7 = 4 + 3 = 2^nd term + 3
• 4^th term = 10 = 7 + 3 = 3^rd term + 3
  and so on.

In general, an arithmetic progression looks like this:

Here,

• 'a' is the first term and
• 'd' is the common difference (fixed number)

Arithmetic Progression Example

For example, Minnie put $30 in her piggy bank when she was 7 years old. She increased the amount she put in her piggy bank on each successive birthday by $3. So, the amount in her piggy bank follows the pattern of $30, $33, $36, and so on. The succeeding terms are obtained by adding a fixed number, that is, $3. This fixed number is called the common difference (It can be positive, negative, or zero). Hence the progression 30, 33, 36, ... is an AP.

Arithmetic Progression Formulas

Let the first term of the progression be a, the common difference be d, and the n^th term be a_n. Then, the arithmetic progression formulas are given by:

• a_n = a + (n - 1) d
• d = a_n - a_n-1
• Sum of the first n terms, S_n = n/2(2a+(n-1)d) (or) S_n = n/2(a + l), where l = the last term = T_n.

A geometric progression (GP) is a sequence of numbers in which each successive term is the product of its preceding term and a fixed number. This fixed number is called the common ratio. For example, 4, 16, 64, 256, ... is a GP as every number is obtained by multiplying a fixed number 4 to its previous term.

• 2^nd term = 16 = 4(4) = 4(1^st term)
• 3^rd term = 64 = 4(16) = 4(2^nd term)
• 4^th term = 256 = 4(64) = 4(3^rd term)
  and so on.

In general, a geometric progression looks like this:

Here,

• 'a' is the first term and
• 'r' is the common ratio (fixed number)

Geometric Progression Example

Consider an example of a geometric progression: 1, 4, 16, 64, ... Observe that 4/1 = 16/4 = 64/16 = ... = 4. All the ratios are same. Hence it is a GP.

Geometric Progression Formulas

Let the first term of the progression be a, the common ratio be r, and the n^th term be a_n. Then, the geometric progression formulas are given by:

• aₙ = ar^{n - 1}
• Sum of the first 'n' terms, Sₙ = a(rⁿ - 1) / (r - 1) when r ≠ 1 and Sₙ = na when r = 1.
• Sum of infinite geometric series, S_∞ = a / (1 - r) when |r| < 1 and S_∞ diverges when |r| ≥ 1.

A harmonic progression is a sequence obtained by taking the reciprocal of the terms of an arithmetic progression. The sequence of natural numbers is an arithmetic progression. So, taking reciprocals of each term we get 1,1/2,1/3,1/4,... This is harmonic progression.

Harmonic Progression Example

When a ball is dropped, the initial height reached by the ball is 1/2 units and after the first impact, the height attained by the ball is 1/4 units. After the second impact, the height attained by the ball is 1/6 units. After the third impact, the height attained by the ball is 1/8 units, and so on. Now the sequence of heights formed by the ball is: 1/2, 1/4, 1/6, 1/8, ...

This sequence is a harmonic progression because the reciprocal of all the terms of this progression form an arithmetic progression.

• Reciprocal of the sequence: 2, 4, 6, 8, ... → AP
• Hence, the sequence: 1/2, 1/4, 1/6, 1/8, ... → HP

Harmonic Progression Formulas

For a harmonic progression 1/a, 1/(a+d), 1/(a+2d), ...

• n^th term, a_n = 1 / (a + (n - 1) d)
• Sum of the first n terms, S_n = 1/d ln [ (2a + (2n - 1) d] / (2a - d) ]

Important Notes on Progression:

• In an arithmetic progression, each successive term is obtained by adding the common difference to its preceding term.
• In a geometric progression, each successive term is obtained by multiplying the common ratio to its preceding term.
• The reciprocal of terms in harmonic progression form an arithmetic progression.

☛ Related Topics:

• Arithmetic Progression Calculator
• Geometric Progression Calculator

What is the Meaning of a Progression?

Progression is a sequence of numbers that show a pattern. For example, 1/2, 1/4, 1/8, 1/16,... is a progression as there is a pattern where each number is obtained by multiplying its previous number by 1/2 and each number can be written as (1/2)ⁿ (where 'n' represents the term's ordinal number).

What are Types of Progressions?

There are three types of progressions: AP (Arithmetic Progression), GP (Geometric Progression), and HP (Harmonic Progression).

What are Formulas of Progression?

Here are the formulas of different types of progressions. Here, a_n represents the n^th term and S_n represents the sum of the first n terms.

Arithmetic Progression:

• a_n = a + (n - 1) d
• S_n = n/2 (2a + (n - 1) d)

Geometric Progression:

• a_n = a r^{n - 1}
• S_n = a (1 - rⁿ) / (1- r)
• S_∞ = a / (1 - r)

Harmonic Progressions:

• a_n = 1 / (a + (n - 1) d)
• S_n = 1/d ln [ (2a + (2n - 1) d] / (2a - d) ]

What is the Difference Between Arithmetic and Geometric Progressions?

In an arithmetic progression (AP), the differences between every two consecutive terms are all same whereas in a geometric progression (GP), the ratios of every two consecutive terms are all same. For more differences, click here.

What is the Difference Between Arithmetic and Harmonic Progressions?

In an arithmetic progression (AP), every term is obtained by adding a fixed number known as common difference (d) to its previous term. The terms of a harmonic progression (HP) are reciprocals of terms of an AP.

How to Find n^th Term of Progression?

Here are the formulas to find the n^th term of different types of progressions:

• AP: a_n = a + (n - 1) d
• GP: a_n = a r^{n - 1}
• HP: a_n = 1 / (a + (n - 1) d)

How to Find the Sum of n Terms of Progression?

Here are the formulas to find the sum of the first n terms of different types of progressions:

• AP: n/2 (2a + (n - 1) d)
• GP: a (1 - rⁿ) / (1- r)
• HP: 1/d ln [ (2a + (2n - 1) d] / (2a - d) ]