103 LearnersLast updated on July 9th, 2025
Progression
A progression, also known as a sequence, is an ordered set of numbers that follows a specific pattern. In such sequences, each term is obtained based on a specific rule related to the previous term. For example, the sequence 3, 6, 9, 12, ... forms a progression because each number is obtained by adding 3 to the one before it. Progressions can have different patterns based on the type they belong to.
What is Progression?
Progressions are a series of numbers that follow a specific pattern. In a progression, each term is determined by applying a particular rule to the previous term. This pattern can often be described using a general formula, called the nth term, usually denoted as aₙ.
For example, in the progression 4, 7, 10, 13, ..., the nth term is given by the formula:
aₙ = 3n + 1.
By substituting different values of n, we get:
When:
- n = 1, the first term: a₁ = 3(1) + 1 = 4
- n = 2, the second term: a₂ = 3(2) + 1 = 7
- n = 3, the third term: a₃ = 3(3) + 1 = 10
and so on.
Difference Between Sequence and Progression.
Arithmetic Progression (AP) is a type of sequence widely discussed in the field of algebra, dealing with number systems and algebraic operations. Apart from arithmetic progression, other common types include geometric progression and harmonic progression. Every progression is a sequence, not every sequence can be considered a progression.
A sequence is an ordered list of numbers that may or may not follow a specific rule. On the other hand, a progression is a type of sequence in which each term follows a definite pattern or rule. Every term in a progression has a specific position and value within the pattern. For every pattern, there is a recurring rule that connects the terms, which is represented by the nth term of the progression.
What are the Types of Progressions?
The three main types of progression are:
- Arithmetic Progression (AP)
- Geometric progression (GP)
- Harmonic Progression (HP)
Let’s now learn their differences with examples:
|
Progression |
Definition |
Example |
|
Arithmetic Progression (AP) |
A sequence where the difference between any two consecutive terms is constant. |
2, 5, 8, 11, ... |
|
Geometric Progression (GP) |
A sequence where the ratio between any two consecutive terms is constant. |
3, 6, 12, 24, ... |
|
Harmonic Progression (HP) |
A sequence where the reciprocals of the terms form an arithmetic progression (AP). |
1, 1/2, 1/3, 1/4, ... |
Real-Life Applications of Progression
Progressions are number patterns that are widely used in numerous fields. From mathematical concepts to everyday life, they have many practical uses. Let’s now learn how different types of progressions are used in real-world situations:
- An Arithmetic Progression (AP) is formed when you save a fixed amount of money each month. This makes it easy to calculate your total savings easily.For example, if you save a fixed amount of $200 every month, the total savings after 5 months will be: $200, $400, $600, $800, $1000…
- We can estimate the growth using a Geometric Progression (GP) if the population grows by the same percentage each year. This helps in forecasting the population changes over time.
- Geometric Progression (GP) is a concept used to calculate internet or mobile data speed, which usually doubles with each upgrade.
For example, 1GB, 2GB, 4GB, 8GB.
Common Mistakes and How to Avoid Them in Progression
Students mostly confuse progression with sequence. While working on progression, few things need to be followed. Few commonly made mistakes are as following -
Mistake 1
Confusion Between AP and GP
Sometimes, students confuse the patterns of AP and GP.
For example, trying to find the common difference in the sequence 2, 4, 8, 16, … by assuming it as an AP.
Ensure that you check the pattern before confirming its type:
- When every term increases or decreases by the same number, it's AP.
- When every term is divided or multiplied by the same number, it forms a GP.
Given that the ratio is ×2, this is a GP.
Mistake 2
Applying the Incorrect Formula for the nth Term
When finding a term in an AP, students mistakenly use the formula Tn = a + nd.
For example, if the first term is 3 and the common difference is 5, the student tries to determine the 10th term using the formula:
T₁₀ = 3 + 10 × 5 = 53.
This is incorrect.
The correct formula is:
Tₙ = a + (n - 1) × d.
So, the correct calculation is:
T₁₀ = 3 + (10 - 1) × 5 = 3 + 45 = 48.
Mistake 3
Errors in Sign Handling
It is common to make sign errors in the common difference or ratio.
For example, for the AP 10, 7, 4, … , students often think that d = +3.
Check the pattern carefully. Here, the numbers are decreasing, so d = –3.
Mistake 4
Applying Finite GP Formula for the Infinite GP
Using the finite GP formula for infinite GP causes errors when calculating the sum.
If the common ratio |r| < 1, use the formula for the sum of an infinite GP:
S = a / (1 – r).
Mistake 5
Overlooking the First Term ‘a’
Ignoring the first term in an AP can lead to an incorrect sequence.
For example, using only the common difference to determine the terms in an AP.
Identifying the first term is an important step in all formulas before solving them.
Solved Examples of Progression
Problem 1
Find the 10th term of an AP where the first term is 2 and the common difference is 3.
29
Explanation
Given:
a = 2, d = 3, n = 10
Here, we use the formula:
Tₙ = a + (n - 1) × d
Substituting the values into the formula:
T₁₀ = 2 + (10 - 1) × 3
T₁₀ = 2 + 9 × 3 = 2 + 27 = 29
Problem 2
Find the 6th term of a GP where the first term is 5 and the common ratio is 2.
160
Explanation
a = 5, r = 2, n = 6
We have the formula:
Tₙ = a × rⁿ⁻¹
Substituting the values into the formula:
T₆ = 5 × 2⁵ = 5 × 32 = 160
Problem 3
Find the sum to infinity of the GP: 8, 4, 2, 1, ...
16
Explanation
a = 8, r = 1/2
Use the formula for infinite GP (only if |r| < 1):
S = a / (1 - r)
Substituting the values into the formula:
S = 8 / (1 - 1/2) = 8 / (1/2) = 16
Problem 4
Find the sum of the first 5 terms of the AP: 4, 7, 10, ...
50
Explanation
First term a = 4
Common difference d = 3
Number of terms n = 5
Using the formula:
Sₙ = n/2 × [2a + (n - 1) × d]
Substituting the values into the formula:
S₅ = 5/2 × [2×4 + (5 - 1)×3]
S₅ = 5/2 × [8 + 12] = 5/2 × 20 = 50
Problem 5
Find the sum of the first 4 terms of the GP: 3, 6, 12, 24, …
45
Explanation
This is a GP where:
First term (a) = 3
Common ratio (r) = 6 ÷ 3 = 2
Number of terms (n) = 4
Using the formula for the sum of the first n terms of a GP:
Sₙ = a × (rⁿ - 1) / (r - 1)
Substituting the values into the formula:
S₄ = 3 × (2⁴ - 1) / (2 – 1)
S₄ = 3 × (16 - 1) / 1 = 3 × 15 = 45
Therefore, the sum of the first 4 terms is 45.
FAQs on Progression
1.What do you mean by a progression in math?
2.Give the formula for the nth term of an AP.
3.Can we apply progressions in real life?
4.How can I determine whether a sequence is GP or AP?
5.Can there be negative numbers in a progression?
6.How does learning Algebra help students in Australia make better decisions in daily life?
7.How can cultural or local activities in Australia support learning Algebra topics such as Progression?
8.How do technology and digital tools in Australia support learning Algebra and Progression?
9.Does learning Algebra support future career opportunities for students in Australia?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
