Arithmetic Progression: Definition, AP Formula, Nth term and Sum

Understanding arithmetic progression is crucial in mathematics as it helps solve various problems more effectively. Whether you are a student looking to enhance your math skills or an enthusiast curious about the beauty of numbers, this guide will provide you with valuable insights into the world of arithmetic progression. Get the formula and explanation in detail. 

What is Arithmetic Progression (AP) in Maths? 

An Arithmetic Progression (AP) in maths is a sequence of numbers (2, 5, 8, 1,....) in a specific order where the difference between two consecutive terms is constant (d=3). This constant difference is known as the common difference.

Each term in an arithmetic progression can be calculated by adding the common difference to the preceding term. The general form of an arithmetic progression can be represented as a_n= a₁ + (n - 1)d, where a_n is the nth term. 

Important Terminologies in Arithmetic Progression

Basic Terms in Arithmetic Progression

Let us study some of the basic terms in arithmetic progression:

Consecutive Terms: In an arithmetic progression, consecutive terms are the terms that follow each other in sequence. The first term plays a crucial role as it sets the starting point for the progression.

Last Term: The last term marks the endpoint of the arithmetic progression. It is determined by the number of terms and the common difference.

Successive Terms: Successive terms in an arithmetic progression are calculated by adding or subtracting the common difference from the preceding term. This process continues until the desired term is reached.

Previous Term: The previous term in an arithmetic progression is always one step behind the current term. It is obtained by either subtracting or adding the common difference to reach back a step.

Differences: The differences between consecutive terms in an arithmetic progression are constant, representing the common difference. This consistent variance forms the foundation of arithmetic progressions.

General Term: The general term of an arithmetic progression describes any term within the sequence using variables such as 'n' for position and 'a' for the first term. It helps generalize patterns in progressions.

What is the First Term in AP? 

In arithmetic progression, the first term is denoted by 'a'. It represents the initial value of the sequence. For instance, if we have a sequence starting with 2, then a = 2. The first term serves as the foundation for the entire sequence and determines all subsequent values. It's crucial in defining the progression and understanding its structure.

What is the Common Difference in AP? 

The common difference between consecutive numbers in an arithmetic progression is represented by 'd'. This value signifies how much each term increases or decreases from the preceding. For example, consider a sequence where each number is 3 more than the previous one. In this case, d = 3. The common difference defines the pattern and relationship between terms.

n-th Term in Arithmetic Progression

In an arithmetic progression, the term at position 'n' is denoted as 'a_n'. This symbol represents a particular term in the sequence.

If we want to find the 5th term in a sequence with a common difference of 4, we use the formula: a₅ = a + (n-1)d, where n = 5.

General Form Of Arithmetic Progression Series

Generally, an arithmetic progression (AP) can be written as (a, a + d, a + 2d, a + 3d, and so on), where 'a' is the starting number and 'd' is the fixed amount added to each term. To represent an arithmetic progression series, you start with the first term denoted by 'a' and add the common difference' d' successively to obtain each subsequent term.

Formula for n-th Term in Arithmetic Progression

The formula to find the n-th term in AP is a_n = a + (n-1)×d, where

• (a_n) represents the nth term,

• 'a' denotes the first term,

• 'd' is the common difference,

• 'n' signifies the position of the term in the sequence.

When finding a specific term in an AP series, you can use this formula by plugging in the values of 'a', 'd', and 'n'.

Summation of the n-th term in AP (Formula)

The sum of the initial 'n' terms in an arithmetic progression is determined by a formula called the sum of an arithmetic series. It is represented as: S_n= n/2 [2a+(n-1)×d]

• (S_n): sum of the first n terms

• (n): number of terms being summed

• (a): the first term of the AP

• (d): the common difference between consecutive terms

Types Of Arithmetic Progression

Let us study the types of arithmetic progression:

Finite AP

Finite arithmetic progressions have a specific number of terms and a well-defined endpoint. The final term in the sequence limits them, making them finite. The common difference between consecutive terms remains constant throughout the progression.

Infinite AP

Infinite arithmetic progressions continue indefinitely without an endpoint. They have an infinite number of terms that follow a specific pattern dictated by the common difference. 

Click here to enhance and upskill your quantitative aptitude by exploring different topics, including arithmetic progression.

Solved Questions With Explanation (MCQs)

Practice is the key to excelling in quantity aptitudes, such as arithmetic progression! Below are some of the top selected practice questions with answers:

Question 1: A sequence is given as 5,9,13,17,…

What is the 15th term of this arithmetic progression?

A) 57
B) 61
C) 65
D) 69

Correct option: B) 61

Explanation: First term, a₁=5

Common difference, d=9−5=4

Number of terms, n=15

Substituting this into the formula a_n=a₁+(n−1)×d, we have, a₁₅=5+(15−1)×4=5+14×4= 5 + 56 = 61.

Question 2: What is the fifth term of the arithmetic progression with a first term of 3 and a common difference of 4?

A) 15
B) 16
C) 19
D) 18

Correct answer: C) 19
Explanation: Same process as in question 1. Apply the formula a_n=a₁+(n−1)×d by substituting these values a₁=3, d=4 & n=5. 

Question 3: What is the sum of the first 10 terms of the arithmetic progression 1, 3, 5, 7, ...?

A) 100

B) 110

C) 90

D) 120

Correct answer: A) 100

Explanation: For the given AP:

• The first term a=1
• The common difference d=3−1=2
• The number of terms n=10

Plugging these values into the formula: S_n=n/2×[2a+(n−1)×d]

S₁₀=10/2×(2×1+(10−1)×2=5×(2+9×2)=5×(2+18)=5×20=100

So, the sum of the first 10 terms of the arithmetic progression is 100.

Question 4: Find the 15th term and the sum of the first 15 terms of the arithmetic progression with the first term a=4 and common difference d=3.

A) 25, 354

B) 35, 324

C) 59, 379

D) 46, 375

Correct answer: D) 46, 375

Explanation: The formula for the $n$-th term of an AP is: a_n=a+(n−1)×d

For the 15th term (n=15), 

a₁₅=4+(15−1)×3=4+14×3=4+42=46

So, the 15th term is 46.

The formula for the sum of the first $n$ terms of an AP is: S_n=n/2×[2a+(n−1)×d]

For n=15,

S₁₅=15/2×[2×4+(15−1)×3]=15/2×(8+14×3)=15/2×(8+42)=15/2×50=15×25= 375

So, the sum of the first 15 terms is 375.

Question 5: What is the 12th term of the arithmetic progression 8, 11, 14, 17, ...?

A) 32

B) 35

C) 38

D) 41

Correct answer: D) 41

Explanation: Same as question 2. Substitute the values accordigly in the formula- a_n=a+(n−1)×d

Question 6: Find the sum of the arithmetic progression 10, 15, 20, ..., 50.

A) 270

B) 300

C) 350

D) 400

Correct answer: A) 270

Explanation: Same as in question no 3. Put the values into the formula: S_n=n/2×[2a+(n−1)×d]

Question 7: If the sum of the first 15 terms of an arithmetic progression is 180 and the common difference is 2, what is the first term?

A) 3

B) -2

C) 6

D) 9

Correct answer B): -2

Explanation: Same as in question no 3. Put the values into the formula: S_n=n/2×[2a+(n−1)×d]

Question 8: In an arithmetic progression, the 9th term is 25, and the 18th term is 52. Find the sum of the first 15 terms.

A) 320

B) 330

C) 340

D) 350

Correct answer: B) 330

Explanation

To solve this, let us first find the Common Difference ($d$)

Using the formula for the $n$-th term of an AP a_n=a+(n−1)×d, we can find the 9th term.

a₉=a+8d=25

For the 18th term, 

a₁₈=a+17d=52

We have two equations:

a+8d=25 (1)

a+17d=52 (2)

Subtract the first equation from the second to eliminate '$a\text{'}$:

(a+17d)−(a+8d)=52−25

a+17d−a−8d=52−25

9d=27

d=3

Secondly, let us find the First Term ($a$)

Substitute d=3 into the first equation,

a+8d=25

⇒a+8×3=25 

 

⇒a=25-24=1

 Finally, we find the sum of the first 15 terms (S₁₅) using the formula S_n=n/2×[2a+(n−1)×d]

For n=15

S₁₅=15/2×[(2×1+(15−1)×3]​=15/2×(2+42)=15/2×44=15×22$=330$

So, the sum of the first 15 terms is 330.

Question 9: The 10th term of an arithmetic progression is 28, and the 16th term is 46. Find the sum of the first 20 terms.

A) 690

B) 700

C) 590

D) 720

Correct answer: C) 590

Explanation: Same process as in question 8. 

Question 10: If the sum of the first n terms of an arithmetic progression is n²+5n, find the first term of the progression.

A) 6

B) 2

C) 3

D) 4

Correct answer: A) 6

Explanation: We are given the sum of the first $n$ terms of an arithmetic progression (AP) as:

S_n=n²+5n

To find the first term ($a$), we use the fact that the sum of the first term is equal to S₁, which is the sum when n=1.

Substitute n=1 into the equation nS_n=n²+5n, we get, 

S₁=1²+5×1$$=6$$

Therefore, the first term $a$ of the progression is 6.

Conclusion

Having delved into the intricacies of Arithmetic Progression (AP), one can appreciate its fundamental role in mathematics. Understanding the notation, formulas, and types of AP series equips individuals with the tools to solve problems efficiently.

The ability to calculate the sum of n-terms and apply AP in various scenarios highlights its practical significance. To master AP, continuous practice and application of the concepts learned are crucial. By honing these skills, one can tackle complex problems confidently and precisely in mathematics.

Frequently Asked Questions (FAQs)

1. What is Arithmetic Progression?

An Arithmetic Progression (AP) is a series of numbers in which each term is determined by adding a consistent value to the preceding term.

2. What does the general form of the AP series refer to?

The general form of an Arithmetic Progression series is represented as (a+d, a+2d, ..., a+(n-1)d), where (a) is the first term, (d) is a common difference, and (n) is the number of terms in the series.

3. Why are key formulas important in Arithmetic Progression?

Key formulas in Arithmetic Progression provide essential tools for calculating various aspects such as individual terms, sum of terms, and identifying patterns within the sequence. They streamline calculations and aid in understanding the properties of AP effectively.

4. What are some applications of Arithmetic Progressions?

Arithmetic Progressions are used in finance, physics, and computer science to model linear relationships, analyze growth patterns, calculate averages, and solve real-world problems with uniform changes.

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