Arithmetic Geometric Progression

Before we understand the meaning of terms like arithmetic geometric progression, let us understand what a passage means. In math and in real life, a person may encounter many examples of progression.

A progression is a series that reveals a particular pattern. In a number system, a passage refers to a sequence of numbers that adhere to a specific pattern. This pattern involves a common difference, a consistent ratio, or other recurring values.

Arithmetic progressions are sequences of numbers where the difference between consecutive terms is constant. This difference is called a 'common difference.' A geometric progression includes multiplying each term by the same factor to obtain the next term, which is called a 'common ratio.'

Read on to find out more about formulas, examples, and differences between arithmetic and geometric progression. 

Understanding Arithmetic Progression

A sequence is an arithmetic progression only if difference between a term and previous term is similar every time.

an + 1 - an = constant (d) for all natural numbers, where an+1 is the term after an. The constant difference is denoted by (d) and is known as the common difference.

For example:

1,5, 9, and 13. Is an A.P. whose first term is 1, and d is equal to 4 (i.e. 5 - 1)

Arithmetic Progression Formulas

When learning about arithmetic progressions, we encounter two significant formulas:

1. The \(n\)th term of an A.P.

2. The sum of the first \(n\) terms of an A.P.

Let's explore both formulas with examples.

The nth Term of an A.P.

The (n)th term of an arithmetic progression is given by:

an = a + (n-1)d

Where:

- (a) is the first term,

- (d) is the common difference,

- (n) is the term number.

The sum of the First (n) Terms of an A.P.

The sum of the first (n) terms of an arithmetic progression is given by:

Sn = n/2 (2a + (n-1)d) or equivalently Sn = n/2 (a +1)

Now, let us understand the types of A.P. in math.

What are the types of A.P. in Math?

Finite A.P.: An arithmetic progression that contains a finite number of terms is called a finite A.P. Such a progression has a definite last term.

Example: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21

Infinite A.P.: An arithmetic progression that does not have a finite number of terms is called an infinite A.P. This type of progression does not have a last term.

Example: 5, 10, 15, 20, 25, 30, 35, 40, 45, …

Now, moving further, let us understand the steps involved in arithmetic progression.

Steps involved in Arithmetic Progression?

To determine if a sequence (an) is an arithmetic progression (A.P.), follow these steps:

Step 1: Identify the general term of the sequence.

Step 2: Replace n with (n+1) in an to find (an+1).

Step 3: Calculate (an+1) - (an).

Step 4: If (an+1 - an) is independent of (n), the sequence is an arithmetic progression.

Otherwise, it is not.

For example: Show that the sequence {an} defined by (an = 4n + 5) is an arithmetic progression. Also, find its common difference.

Solution:

1. Given an = 4n + 5.
2. Replace (n) with (n+1):

            an+1 = 4(n+1) + 5 = 4n + 4 + 5 = 4n + 9

3. Calculate the difference:

           a{n+1} - an = (4n + 9) - (4n + 5) = 4

4. Since (an+1 - an) is 4, independent of (n), the sequence is an arithmetic progression with a common difference of 4.

What is Geometric Progression?

A sequence is a geometric progression (G.P.) if the ratio between a term and its previous term is always constant.

𝑎𝑛+1/𝑎𝑛 = constant (r) for all natural numbers, where 𝑎𝑛 + 1 is the term after 𝑎𝑛

​The constant ratio is denoted by 𝑟 and is known as the standard ratio.

For example:

1, 3, 9, 27, ... is a G.P. with the first term one and a standard ratio of 3 (i.e., 3/1 = 9/3 = 27/9 = 3)

Now, let us understand the geometric progression formula.

Geometric Progression Formulas

The primary and general form of a Geometric Progression is:

a, ar, ar2, ar3, ar4,...., ar (n - 1) in which

a = First term

r = standard ratio and

a (n -1) = nth term

nth Term of a G.P.

The nth term of a geometric progression is given by:

𝑎𝑛 = 𝑎𝑟(𝑛−1)

Where:

𝑎 is the first term,

𝑟 is the standard ratio,

𝑛 is the term number.

The sum of the First (n) Terms of a G.P.

The sum of the first 𝑛 terms of a geometric progression is given by:

𝑆𝑛 = 𝑎 (1−𝑟𝑛) / (1-r) if 𝑟 ≠ 1

Types of G.P. in Math

Finite G.P.: A geometric progression that contains a finite number of terms is called a finite G.P. Such a progression has a definite last term.

Example: 2, 4, 8, 16, 32

Infinite G.P.: A geometric progression that does not have a finite number of terms is called an infinite G.P. This type of progression does not have a last term.

Example: 3, 9, 27, 81

Now, moving further, let us understand the steps involved in arithmetic progression.

Steps involved in Geometric Progression

To determine if a sequence {𝑎𝑛} is a geometric progression (G.P.), follow these steps:

Step 1: Identify the general term 𝑎𝑛 of the sequence.

Step 2: Replace 𝑛 with 𝑛+1 in 𝑎𝑛 to find 𝑎𝑛+1

Step 3: Calculate the ratio 𝑎𝑛 + 1 / 𝑎𝑛

Step 4: If 𝑎𝑛 + 1 / 𝑎𝑛 is independent of n, the sequence is a geometric progression. Otherwise, it is not.

For example: Show the sequence {𝑎𝑛} defined by 𝑎𝑛 = 2 ⋅3𝑛 is a geometric progression. Also, find its standard ratio.

Solution:

Given 𝑎𝑛 = 2 ⋅3𝑛

Replace 𝑛 with n+1:

𝑎𝑛+1 = 2 ⋅3 𝑛+1 = 2⋅3n ⋅3

Calculate the ratio:

𝑎𝑛 + 1 / 𝑎𝑛 = 2 ⋅3𝑛+1 / 2⋅3𝑛 = 3

Since 𝑎𝑛 + 1 / an = 3, which is independent of n, the sequence is a geometric progression with a standard ratio of 3

Now, let us understand the difference between arithmetic progression and geometric progression.

Arithmetic Progression and Geometric Progression: Differences

Now, let us understand a few Practical uses of arithmetic and geometric progression.

Practical Uses of Arithmetic and Geometric Progression

Arithmetic and geometric progressions are not just theoretical concepts but are crucial in real-world scenarios across various industries:

• Finance: Geometric progressions calculate compound interest, showing how investments grow exponentially. Arithmetic progressions are used in loan repayments, such as mortgages, where the mix of interest and principal payments changes predictably each month.
• Engineering: Engineers use geometric progressions for designs that need to scale up exponentially, like certain architectural features. Arithmetic progressions help plan the gradual increase of project resources, ensuring efficient use over time.
• Population Growth: Geometric progressions model population changes, showing how numbers can increase or decrease rapidly under certain conditions.
• Convergence: In geometric progressions, if the standard ratio's absolute value is less than 1, the sequence will eventually converge to zero. This concept helps predict long-term outcomes in finance and environmental studies.

Wrapping Up

This concludes our exploration of arithmetic and geometric progressions. Throughout this tutorial, you've gained insight into calculating the nth term and the sum of n terms in any given series.

Arithmetic and geometric progressions are foundational concepts in statistics, particularly in data analytics. Mastering these concepts is crucial for anyone aspiring to work in data analysis.

Frequently Asked Questions

Let’s take a look at some arithmetic progression and geometric progression questions: 

1. When Does a Geometric Progression Converge?

A geometric progression coincides if the absolute value of the common ratio |r| is less than 1. In such cases, as the number of terms increases, the terms of the progression approach zero, and the sum of infinite geometric progression can be calculated using the formula:

S∞ = a / 1-r

2. What's the Difference Between Arithmetic and Geometric Sequences?

The main difference between arithmetic and geometric sequences lies in how the terms are generated:

• Arithmetic Sequence (A.P.): Each term is acheived by adding a fixed number (the common difference) to the previous term.

• Geometric Sequence (G.P.): Each term is gained by multiplying the previous term by a fixed number (the common ratio).

3. What are the Advantages of Geometric Progression?

• Geometric progressions often arise in natural phenomena, such as population growth, financial investments, and exponential decay.
• They offer a simple and efficient way to model exponential growth or decay processes.
• Geometric progressions are widely used in various fields, including mathematics, finance, physics, and engineering.

4. What is the Common Ratio of a Geometric Progression (G.P.)?

The common ratio of a geometric progression is the fixed number by which each term is multiplied to obtain the next term. It is denoted by \(r\). For example, in the sequence 2, 6, 18, 54, ..., the common ratio is 3.