Arithmetic Progression(AP)-Common difference & Nth term, Practice Problems, FAQs

📢 Master Arithmetic Progression (AP) – Class 11 Mathematics 📚

🚀 Excel in Board Exams & JEE with Neeraj Anand’s Expert Guide! 🚀

Are you preparing for Class 11 Board Exams, JEE Mains & Advanced? Arithmetic Progression (AP) is a fundamental topic in Algebra and a key concept in various real-world applications, making it essential for competitive exams.

🔢 What is Arithmetic Progression (AP)?

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms remains constant. This fixed difference is called the common difference (d).

📌 General Form of an AP : a,a+d,a+2d,a+3d,………..a+(n-1)d

Where:

• a = First term
• d = Common difference
• n = Number of terms

📌 Examples of AP:
1️⃣ Simple AP: 3,7,11,15,19,… (Here, d=4d = 4d=4)
2️⃣ Negative AP: 10,7,4,1,−2,… (Here, d=−3d = -3d=−3)
3️⃣ Real-life Example: Steps in a staircase where each step increases by the same height.

📌 Applications of AP in JEE & Board Exams:
✅ Time and distance problems
✅ Interest calculations in finance
✅ Physics problems involving motion

What is the First Term of an AP ?

The AP can also be written in terms of common differences, as follows;

a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d

where  “a” is the first term of the progression. 

What is the Common difference of an AP?

The common difference in the arithmetic progression is denoted by d. The difference between the successive term and its preceding term. It is always constant or the same for arithmetic progression. In other words, we can say that, in a given sequence if the common difference is constant or the same then we can say that the given sequence is in Arithmetic Progression.

The common difference “ d ” can be obtained as;

d = a2 – a1 = a3 – a2 = ……. = an – an – 1

Where “d” is a common difference. It can be positive, negative or zero.

• The formula to find common difference is d = (a_{n + 1} – a_n ) or d = (a_n – a_n-1).
• If the common difference is positive, then AP increases. For Example 4, 8, 12, 16….. in these series, AP increases
• If the common difference is negative then AP decreases. For Example -4, -6, -8……., here  AP decreases.
• If the common difference is zero then AP will be constant. For Example 1, 2, 3, 4, 5………, here AP is constant.

The sequence of Arithmetic Progression will be like a₁, a_2, a₃, a₄,…

Examples of AP

Example 1: 0, 5, 10, 15, 20…..

here, 
a1 = 0, a2 = 5, so a2 - a1 = d = 5 - 0 = 5. 
a3 = 10, a2 = 5, so a3 - a2 = 10 - 5 = 5.
a4 = 15, a3 = 10, so a4 - a3 = 15 - 10 =5.
a5 =20, a4 =15, so a5 -a4 = 20 - 15 = 5.

From the above example, we can say that the common difference is “5”.

Example 2: 0, 7, 14, 21, 28…….

here, 
a1 = 0, a2 = 7, so a2 - a1 = 7 - 0 = 7
a3 = 14, a2 = 7, so a3 - a2 = 14 - 7 = 7
a4 = 21, a3 = 14, so a4 - a3 = 21 - 14 = 7
a5 =28, a4 = 21, so a5 -a4 =28 - 21 = 7

From the above example, we can say that the common difference is “7”.

How to Find the Middle term of an AP?

To find the middle term of an arithmetic progression we need the total number of terms in a  sequence. We have two cases:

Even: If the number of terms in the sequence is even then we will be having two middle terms i.e (n/2) and  (n/2 + 1).n

Odd: If the number of terms in the sequence is odd then we will be having only one middle terms i.e (n/2).

Example 1:

If n = 9 then,
Middle term = n/2 = 9/2 = 4.

Example 2:

If n = 16 then,
First middle term = n/2 = 16/2 = 8.
Second middle term = (n/2) + 1 = (16/2) + 1 = 8 + 1 = 9.

General Form of an AP

Consider an AP to be: a₁, a₂, a₃, ……………., a_n

Position of Terms	Representation of Terms	Values of Term
1	a1	a = a + (1-1) d
2	a2	a + d = a + (2-1) d
3	a3	a + 2d = a + (3-1) d
4	a4	a + 3d = a + (4-1) d
.	.	.
.	.	.
.	.	.
.	.	.
n	an	a + (n-1)d

What is the Nth term of an AP?

To find the nth term of an arithmetic progression, We know that the A.P series is in the form of a, a + d, a + 2d, a + 3d, a + 4d……….a+(n-1)d

The n^th  term is denoted by  T_n. Thus to find the nth term or last term of an A.P series will be : 

T_n = a+(n-1)d

Example: Find the 9^th term of the given A.P sequence: 3, 6, 9, 12, 15………..?

Step 1: Write the given series.

Given series = 3, 6, 9, 12, 15...........

Step 2: Now write down the value of a  and n from the given series.

a = 3, n = 9

Step 3: Find the common difference d by using the formula  (a_n+1  – a_n).

d = a2 - a1 , 
here  a2 = 6 and a1 = 3  
so d = (6 - 3) = 3.

Step 4: We need to substitute values of a, d, n  in the formula (T_n = a + (n – 1)d).

Tn =  a + (n - 1)d 
given n = 9.

T9 = 3 + (9 - 1)3

= 3 + (8)3

= 3 + 24 = 27

Therefore the 9^th  term of given A.P series  3, 6, 9, 12, 15………. is “27”.

Types of AP

Finite AP: An AP containing a finite number of terms is called finite AP. A finite AP has a last term.

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

Infinite AP: An AP which does not have a finite number of terms is called infinite AP. Such APs do not have a last term.

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

Arithmetic Progressions Solved Examples

Below are the problems to find the nth term and the sum of the sequence, which are solved using AP sum formulas in detail. Go through them once and solve the practice problems to excel in your skills.

Example 1: Find the value of n, if a = 10, d = 5, a_n = 95.

Solution: Given, a = 10, d = 5, a_n = 95

From the formula of general term, we have:

a_n = a + (n − 1) × d

95 = 10 + (n − 1) × 5

(n − 1) × 5 = 95 – 10 = 85

(n − 1) = 85/ 5

(n − 1) = 17

n = 17 + 1

n = 18

Example 2: Find the 20th term for the given AP:3, 5, 7, 9, ……

Solution: Given, 

3, 5, 7, 9, ……

a = 3, d = 5 – 3 = 2, n = 20

a_n = a + (n − 1) × d

a₂₀ = 3 + (20 − 1) × 2

a₂₀ = 3 + 38

⇒a₂₀ = 41

Example 3: Find the sum of the first 30 multiples of 4.

Solution:

The first 30 multiples of 4 are: 4, 8, 12, ….., 120

Here, a = 4, n = 30, d = 4

We know,

S₃₀ = n/2 [2a + (n − 1) × d]

S₃₀ = 30/2[2 (4) + (30 − 1) × 4]

S₃₀ = 15[8 + 116]

S₃₀ = 1860

Practice Problems on AP

Find the below questions based on Arithmetic sequence formulas and solve them for good practice.

Question 1: Find the a_n and 10th term of the progression: 3, 1, 17, 24, ……

Question 2: If a = 2, d = 3 and n = 90. Find a_n  and S_n.

Question 3: The 7th term and 10th terms of an AP are 12 and 25. Find the 12th term.

Frequently Asked Questions (FAQs) on Arithmetic Progression (AP)

Q1

What is the general form of Arithmetic Progression?

The general form of arithmetic progression is given by a, a + d, a + 2d, a + 3d, . . .. Hence, the formula to find the nth term is:
a_n = a + (n – 1) × d

Q2

What is arithmetic progression? Give an example.

A sequence of numbers that has a fixed common difference between any two consecutive numbers is called an arithmetic progression (A.P.). The example of A.P. is 3,6,9,12,15,18,21, …

Q3

How to find the sum of arithmetic progression?

To find the sum of arithmetic progression, we have to know the first term, the number of terms and the common difference. Then use the formula given below:
S_n = n/2[2a + (n − 1) × d]

Q4

What are the types of progressions in Maths?

There are three types of progressions in Maths. They are:
Arithmetic Progression (AP)
Geometric Progression (GP)
Harmonic Progression (HP)

Q5

What is the use of Arithmetic Progression?

An arithmetic progression is a series which has consecutive terms having a common difference between the terms as a constant value. It is used to generalise a set of patterns, that we observe in our day to day life. For example, AP used in prediction of any sequence like when someone is waiting for a cab. Assuming that the traffic is moving at a constant speed he/she can predict when the next cab will come.

🔢 What’s Inside This Chapter?
✅ Definition & Basics of AP – Understanding Sequences
✅ Common Difference (d) – Identifying the Pattern
✅ nth Term Formula – Finding Any Term in an AP
✅ Sum of n Terms (Sn) Formula – Quick Calculation Techniques
✅ Applications in Real Life & JEE Problems – Mastering AP for Competitive Exams

📖 Why Choose This Book?
✔️ Step-by-Step Explanations for Conceptual Clarity
✔️ Solved Examples & Extensive Practice Questions
✔️ Special Focus on JEE-Level Problems & Shortcuts

🔗 📥 Download the PDF & Start Learning Now!

📕 Written by: Neeraj Anand
🏛 Published by: ANAND TECHNICAL PUBLISHERS
🏫 Available at: ANAND CLASSES

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