What is the difference between Arithmetic Progression and Geometric Progression?

In modern mathematics, number theory is used, which is basically doing all operations in numbers, and the most basic method of solving decimal number systems in number theory is arithmetic. Arithmetic deals with calculations like addition, subtraction, multiplication, exponential, etc. There are sequences and progressions in number theory which are basically numbers aligned in a manner that they offer a certain relationship. Let’s take a look at what progressions are,

Progression

Progression can be defined as a list of numbers having a particular pattern. The next number in a progression can be calculated using a formula. The relation between numbers must be the same always and relevant. Progressions are mainly used to identify the nth term in the sequence. Based on the type of relation between numbers progressions are divided into three types.

1. Arithmetic Progression
2. Geometric Progression 
3. Harmonic Progression.

In this article let’s discuss Arithmetic and Geometric progressions. 

What are Arithmetic progression and Geometric progression?

Arithmetic and geometric progression come under the topic of sequence and series. Arithmetic series/ progression is basically the progression where the difference between two consecutive numbers is always the same, known as a Common difference, denoted by “d”. Geometric progression is the one where the ratio of two consecutive numbers is always the same, noted by “r”. Let’s take a deeper look into these,

Arithmetic Progression (AP)

Arithmetic Progression can be defined as a sequence of numbers in which two consecutive terms in the series will have the same common difference or a sequence of numbers in which exactly the same number is added to each term as the progression continues. Example: 2, 4, 6, 8, 10 this series is in Arithmetic progression because each consecutive term is obtained by adding some constant number 2 to the before term. 

Terms related to Arithmetic Progression

There are few main terms that are related to Arithmetic Progression that we come across frequently they are:

First term (a) – The very first term in the progression is represented as ‘a’. First-term in the progression is used to calculate the nth term and the sum of numbers in the progression. Progression is represented as

a, a+d, a+2d, a+3d, a+4d,…n terms

Common Difference (d) – Common difference is defined as the difference between two consecutive terms. This is represented as d and can be obtained as 

d = a₂-a₁ = a₃-a₂ = a₄-a₃ =…..= a_n-a_n-1 

n^th term (a_n) – The nth term is defined as the term that comes at the nth place in the sequence. It is represented as a_n obtained as

a_n = a + (n-1)d

where,

a – First term in the progression

n – number of terms

d – common difference

Question: Calculate 27^th term from the Arithmetic Progression 3, 7, 11, 15, 19, 23,…?

Solution:

a = 3

common difference (d) = a₂ – a₁ = 7 – 3 = 4

Since, nth term in the progression is given by formula

a_n = a + (n-1)d

here n = 27

a₂₇ = 3 + (27-1)(4)

a₂₇ = 3 + (26)(4)

a₂₇ = 107

Therefore 27th term in the progression is 107.

Sum of first n terms (S_n): The sum of n terms is defined as the sum of first n elements in the progression. It is denoted by S_n and can be obtained by using the formula,

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

where,

a – First term in the progression

n – number of terms

d – common difference

Question: Calculate the sum of the first 100 terms in the Arithmetic Progression 3, 7, 11, 15, 19, 23,….?

Solution: 

a = 3

common difference (d) = a₂ – a₁ = 7 – 3 = 4

Since, sum of first n terms in a progression is given by formula 

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

Here n = 100

S₁₀₀ = (100/2)(2× 3 + (100-1)× 4)

S₁₀₀ = (50)(6 + 396)

S₁₀₀ = 20100

Therefore sum of first 100 terms in the given progression is 20100

Geometric Progression (GP)

Geometric Progression is defined as a sequence of terms in which any two consecutive has a common ratio. In this sequence, the next term is obtained by multiplying a constant term to the previous term and the previous term can be obtained by dividing a constant term into the term. Example: 2, 6, 18, 54, 162, 486, 1458,… This sequence is in Geometric Progression with a constant ratio of 3. When consecutive terms are divided we get a constant term 3 which is called a common ratio.

There are two types of Geometric Progression, Finite and Infinite Progression,

Finite Geometric Progression: This type of progression has a finite number of terms and the last term is defined

Infinite Geometric Progression: This type of Geometric Progression has an infinite number of terms and the last term is not defined.

Terms related to Geometric Progression

First term (a): The very first term in the progression is represented as ‘a’. First-term in the progression is used to calculate the nth term and the sum of numbers in the progression. Progression is represented as

a, ar, ar², ar³,…. n terms

Common ratio (r): Common ratio in the progression can be obtained by dividing any two consecutive terms in the progression. It is represented by ‘r’ and can be positive or negative.

Common ratio = (term) / (Previous term) = a₂/a₁ = a₃/a₂ = a_n/a_n-1

nth term (a_n): the nth term is defined as the term that comes at the nth place in the sequence. It is represented as an obtained as

a_n = ar^n-1

where,

a – First term

r – Common ratio

n – Number of terms

Question: Calculate 16th term in the given Geometric Progression 3, 9, 27, 81,………..?

Solution: 

a = 3

Common ratio (r) = a₂/a₁ = 9/3 = 3

Since, nth term in Geometric Progression is given by formula

a_n = ar^n-1

Here n = 16

a₁₆ = 3× (3)^16-1

a₁₆ = 43,046,721

Therefore 16th term in the progression is 43046721

4. Sum of first n terms (Sn): Sum of n terms is defined as the sum of first n elements in the progression. It is denoted by Sn and can be obtained by using the formula,

Finite Geometric Progression:

S_n = a(1-rⁿ)/(1-r) when r<1

S_n = a(rⁿ-1)/(r-1) when r>1

Infinite Geometric Progression:

S_n = a/(1-r), when |r|<1

where,

a – First term

r – Common ratio

n – Number of terms

Sample Questions

Question 1: Check if the sequence 5, 10, 15, 20, 25, 30,…is in Arithmetic Progression or not?

Solution:

a = 5

To check the given sequence is in Arithmetic Progression the sequence must follow condition,

a₂– a₁ = a₃-a₂

10-5 = 15-10

5 = 5

Since, the condition is satisfied the above sequence is in Arithmetic Progression.

Question 2: Calculate 20th term from the Arithmetic Progression 2, 6, 10, 14, 18, 22,….?

Solution: 

a = 2

common difference (d) = a₂ – a₁ = 6 – 2 = 4

Since, nth term in the progression is given by formula

a_n = a + (n-1)d

here n = 20

a₂₀ = 2 + (20-1)(4)

a₂₀ = 2 + (19)(4)

a₂₀ = 78

Therefore 20th term in the progression is 78.

Question 3: Calculate 10th term in the given Geometric Progression 2, 8, 32, 128,….?

Solution: 

a = 2

Common ratio (r) = a₂/a₁ = 8/2 = 4

Since, nth term in Geometric Progression is given by formula

a_n = ar^n-1

Here n = 10

a₁₀ = 2× (4)^10-1

a₁₀ = 524288

Therefore 10th term in the progression is 524288.

Question 4: Check if the sequence 5, 20, 80, 320….is in Geometric Progression or not?

Solution:

a = 5

To check the given sequence is in Geometric Progression the sequence must follow condition,

a₂/a₁ = a₃/a₂

20/5 = 80/20

4 = 4

Since, the condition is satisfied the above sequence is in Geometric Progression.

Question 5: Calculate the sum of the first 10 terms in the Arithmetic Progression 2, 6, 10, 14, 18, 22,…?

Solution: 

a = 2

common difference (d) = a₂ – a₁ = 6 – 2 = 4

Since, sum of first n terms in a progression is given by formula 

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

Here n = 10

S₁₀ = (10/2)(2*2 + (10-1)*4)

S₁₀ = (5)(4 + 36)

S₁₀ = 200

Therefore sum of first 10 terms in the given progression is 200

Question 6: Calculate the sum of the first 5 terms in the Geometric Progression 128, 32, 8, 2,………..?

Solution: 

a = 128

Common ratio (r) = a₂/a₁ = 32/128 = 1/4 = 0.25

Since the sum of first n terms in an Infinite Geometric Progression is given by the formula,

S_n = a/(1-r), when |r|<1

Here n = 5

S₅ = 128/(1-0.25)

S₅ = 170.6

Therefore the sum of the first 5 terms in the given progression is 170.6

Question 7: Calculate the sum of the first 7 terms in the Geometric Progression 5, 20, 80, 320…20480?

Solution: 

a = 5

Common ratio (r) = a₂/a₁ = 20/5 = 4

Since, sum of first n terms in an Finite Geometric Progression is given by formula

Sn = a(rⁿ-1)/(r-1) r>1

Here n = 7

S₇ = (5) (4⁷-1)/(4-1)

S₇ =  27305

Therefore sum of first 7 terms in the given progression is 27305