Arithmetic Progression: Formulas, Nth Term, Key Terms, Solved Examples
Arithmetic Progression is a sequence of numbers with a constant difference between consecutive terms. Learn about Arithmetic Progression formulas, examples, and real-world applications here.
Chandni 18 Dec, 2024
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Arithmetic Progression (AP) is one of the simplest and most useful concepts in mathematics. It’s a sequence of numbers where the difference between consecutive terms is always constant.
This predictable pattern makes AP valuable in solving problems in everyday life and various fields like finance, construction, physics, and computer science.
For example, it’s used to calculate loan repayments, design staircases, and analyze uniform motion. AP isn’t just about numbers—it’s about understanding patterns and using them to simplify complex problems. In this blog, we will learn the concept of Arithmetic Progression in detail.
What is an Arithmetic Progression?
An Arithmetic Progression (AP) is a sequence of numbers where each term is obtained by adding (or subtracting) a fixed value to the previous term. This fixed value is known as the common difference (denoted as d ).Arithmetic Progressions Examples
Increasing Sequence :
The sequence 3, 7, 11, 15, 19, ... is an AP where the common difference is 4 (each term increases by 4).Decreasing Sequence :
The sequence 20, 15, 10, 5, 0, ... is an AP with a common difference of -5 (each term decreases by 5).Fractional Sequence :
The sequence 1/2, 1, 3/2, 2, 5/2, ... is an AP with a common difference of 1/2 .Negative Sequence :
The sequence -1, -4, -7, -10, -13, ... is an AP with a common difference of -3 . In each of these examples, the difference between consecutive terms remains consistent, making them all arithmetic progressions.General Form of an Arithmetic Progression
The general form of an AP can be written as: a,a+d,a+2d,a+3d,...,a+(n−1)d Where:- a = First term of the sequence
- d = Common difference
- n = Position of the term in the sequence
Key Terms in Arithmetic Progression
First Term (a)
The initial number in the sequence.Example : In the sequence 5, 8, 11, 14 , the first term is 5 .
Common Difference (d)
The fixed value is added or subtracted to each term to get the next term. Example: In the sequence 3, 7, 11, 15 , the common difference is 4 .Nth Term (aₙ)
The term is located at the nth position of the sequence. It can be calculated using the formula: aₙ = a + (n - 1)dExample : In the sequence 2, 5, 8, 11 , the 4th term is 11 .
Sum of First n Terms (Sₙ) :
The total sum of the first n terms of the sequence. The formula is: Sₙ =n/2(2a+(n−1)d) Example: The sum of the first 4 terms of the sequence 1, 3, 5, 7 is 16 .Nth Term Formula
The formula to find the nth term of an arithmetic progression (AP) is:a n =a+(n−1)×d
Where:- aₙ = nth term
- a = First term
- d = Common difference
- n = Position of the term
Example:
Find the 12th term of the AP: 4, 9, 14, 19, ...- First term ( a ) = 4
- Common difference ( d ) = 9 - 4 = 5
- n = 12
Sum of First n Terms (Sₙ) Formula
The formula to calculate the sum of the first n terms of an AP is: S n =n/2×(2a+(n−1)×d) Alternatively, if the last term ( l ) is known: S n =n/2×(a+l)Example
Find the sum of the first 20 terms of the AP: 3, 8, 13, 18, ...- First term ( a ) = 3
- Common difference ( d ) = 8 - 3 = 5
- n = 20
Using the formula:
S 20 = 20/2 × ( 2 × 3 + ( 20 − 1 ) × 5 ) = 10 × ( 6 + 95 ) = 10 × 101 = 1010 The sum of the first 20 terms is 1010 .Types of Arithmetic Progression
Arithmetic progressions are a foundational concept in mathematics, defined by a constant difference between consecutive terms in a sequence. These progressions are categorized into two primary types, each with unique characteristics:Finite Arithmetic Progression
A finite arithmetic progression is a sequence that has a fixed number of terms and comes to an end. This type is commonly encountered in real-life scenarios like measuring quantities or setting fixed intervals.Example: 2, 4, 6, 8, 10 (a total of 5 terms).
Infinite Arithmetic Progression
An infinite arithmetic progression is a sequence that continues indefinitely, with no defined endpoint. These are often used in theoretical problems and mathematical modeling.Example: 1, 2, 3, 4, 5, ... (continues endlessly).
Symmetry
In a finite AP, the sum of terms equidistant from the start and end is always the same.Example: In the AP 2, 4, 6, 8, 10:
The sum of the 1st and 5th terms (2 + 10) equals the sum of the 2nd and 4th terms (4 + 8). This symmetry provides a useful shortcut when analyzing or calculating the sum of terms in a finite sequence.Constant Common Difference
The defining feature of an AP is the constant difference between consecutive terms. This common difference remains the same throughout the sequence, making APs predictable and easy to work with.Example: In the AP 3, 6, 9, 12, the common difference is 3.
Middle Term in Odd Number of Terms
When an AP has an odd number of terms, the middle term is equal to the average of the first and last terms.Example: In the AP 1, 3, 5, 7, 9:
The middle term is 5, calculated as the average of 1 and 9: 1+9/ 2=5 This property simplifies calculations and provides insight into the sequence’s structure.Arithmetic Mean
The arithmetic mean of two numbers is another key concept linked to AP. For any two numbers aa and bb , the arithmetic mean is: Arithmetic Mean= (a+b)/2Example: For the numbers 4 and 10, the arithmetic mean is:
(4+10)/2 = 7 This mean can also represent the middle term of an AP formed with aa and bb as the first and last terms.Real-World Applications of Arithmetic Progression
Arithmetic Progression (AP) is widely used in various fields, making calculations easier and more structured. Here are some key applications:- Banking and Finance : AP helps calculate loan repayments, interest rates, and savings growth over time.
- Construction : Used in designing steps, ramps, and structures with fixed measurements.
- Sports and Gaming : Scoring patterns in sports or game levels often follow AP.
- Computer Algorithms : AP is used in loop iterations, search algorithms, and time complexity calculations.
- Physics : AP models uniform motion, where an object moves at a constant speed.
| Related Articles | |
| Ordinal Numbers | Area of Rectangle |
| Natural Numbers | Even Numbers |
| Angles | Multiplication Sums |
Arithmetic Progression FAQs
Can an Arithmetic Progression have negative numbers?
Yes, an AP can include negative numbers if the first term or common difference leads to them.
Example: 5, 2, -1, -4 (common difference = -3).
What happens if the common difference in an AP is zero?
All terms of the sequence will be the same.
Example: 7, 7, 7, 7 (common difference = 0)
What’s the difference between finite and infinite APs?
Finite APs have a set number of terms, while infinite APs go on forever.
Example: Finite: 1, 3, 5, 7 (4 terms). Infinite: 2, 4, 6, 8, ..
Does AP always increase?
No, an AP can increase, decrease, or remain constant depending on the common difference.
Can AP terms include decimals or fractions?
Absolutely! AP terms can be any numbers, including decimals or fractions.
Example: 1.5, 2.5, 3.5 (common difference = 1).
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