Nth Term of Arithmetic Sequence(AP) Formula | Solved Examples, Practice Problems, FAQs

📢 Master the nth Term of an Arithmetic Progression (AP) – Class 11 Mathematics 📚

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Are you preparing for Class 11 Board Exams, JEE Mains & Advanced? Understanding the nth term of an Arithmetic Progression (AP) is essential for solving sequence-based problems quickly and efficiently.

🔢 Key Concepts Covered in This Chapter:
✅ Definition of Arithmetic Progression (AP)
✅ Formula for the nth Term

✅ Finding Any Term Without Listing All Terms
✅ Applications in Real-Life & JEE Problems

How to find the nth term of an Arithmetic Sequence?

Use the formula: a_n = a₁ + (n – 1)d

Where:
a_n = nth term,
a = first term,
d = common difference,
n = term number.

Substitute the values of a, d, and n into the formula to calculate a_n.

Proof : Assume that a₁, a₂, a₃,… be an arithmetic progression (AP), in which first term a₁ is equal to “a” and the common difference is taken as “d”, then the second term, third term, etc can be calculated as follows:

Second term, a₂ = a+d

Third term, a₃ = (a+d)+d = a+2d,

Fourth term, a₄ = (a+2d)+d = a+3d, and so on.

Therefore, the nth term of an AP (a_n) with the first term “a” and common difference “d” is given by the formula:

n^th term of an AP, a_n = a+(n-1)d.

(Note: The nth term of an AP (a_n) is sometimes called the general term of an AP, and also the last term in a sequence is sometimes denoted by “l”.)

Steps to find the n^th Term of an Arithmetic Sequence

Step 1: Identify the First and Second Term:

• 1st and 2nd term, that is a₁ and a₂.

Step 2: Find the Common Difference:

• Subtract the first term from the second term (d = a₂ – a₁)

Step 3: Use the Formula:

a_n = a₁ + (n – 1)d

Step 4: Substitute Values:

• Plug in the values of a, d, and n into the formula.

Step 5: Simplify and Solve:

• Perform the arithmetic operations to find the nth term.

Arithmetic Sequence (AP) Formula Solved Problems

Question 1: Find the 16^th term in arithmetic sequence 0, 2, 4, 6, 8, 10, 12, 14…..
Solution:

The given arithmetic sequence is:
0, 2, 4, 6, 8, 10, 12, 14, …..
nth term formula is:
an = a₁ + (n – 1)d
From the given,

a₁ = 0 ;

n = 16 ;

d = 2

a₁₆ = 0 + (16 – 1)2

a₁₆ = 15 × 2

a₁₆ = 30

Question 2: Find the 9th term of the given series, 1, 4, 7, 10, 13, 16,….

Solution:

Term number (n) is 9,

1st-term, a₁ = 1 
2nd term, a₂ = 4,

Now find the common difference, 
d = a₂ – a₁ = 4 – 1 = 3

Now the 9th term, 

a₉ = First term + (Last term – 1) × common difference 
= a₁ + (n – 1)d
= 1 + (9 – 1) × 3
= 1 + 8 × 3
= 1 + 24
= 25 

So, the 9th term is 25.

Question 3: What is the 25th term of the arithmetic sequence 21, 15, 9, 3, ….?

Solution:
Given the arithmetic sequence is:
21, 15, 9, 3,…
Here, a₁ = 21
d = a₂ – a₁ = 15 – 21 = -6
nth term
a_n = a₁ + (n – 1)d
25th term of the given sequence is:
a₂₅ = a₁ + (25 – 1)d
= 21 + 24(-6)
= 21 – 144
= -123

Question 4: Find the 7th term of an AP whose 3rd term is 9 and 5th term is 15?

Solution:

Given

• 3rd term (a₃) = 9
• 5th term (a₅ ) = 15

Here we have to find common difference and first term(a₁).

a₃ = a₁ + 2d = 9  [from the formula] ⇢  (1)

And, a₅ = a₁ + 4d = 15  ⇢  (2)

Solve (1) and (2),

a₁ + 2d = 9 ⇢  (1)
a₁ + 4d = 15  ⇢  (2)

Let’s apply subtraction between (1) and (2) 
2d = 6, d=3

So, the common difference is 3

Now put the value of d in any one equation, here put the value of d in (1)
a₁ + 2d = 9
= a₁ + 2 × 3 = 9    [d = 3]
= a₁ = 9 – 6
= a₁ = 6

So the first term is 3

Now to find the 7th term,

Apply the formula for finding the nth term, here n = 7
a₇ = First term + (7th term – 1) × common difference
a₇ = a₁ + (7 – 1)d
a₇ = 3 + 6 × 3   [d = 3 and a₁ = 3]
a₇ = 21

So the 7th term is 21.

Question 5: Determine the 10th term of an AP 2, 7, 12, ….

Solution:

Given arithmetic progression (AP) is 2, 7, 12, …

Here, the first term, a = 2.

Common difference, d = 7-2 = 5

n=10.

The formula to find the nth term of an AP, a_n = a+(n-1)d

Now, substitute the values in the formula, we get

a₁₀ = 2 + (10-1)5

a₁₀ = 2 + (9)5

a₁₀ = 2+45

a₁₀ = 47.

Hence, the 10th term of an AP 2, 7, 12, … is 47.

Question 6 : The third term of an AP is 5 and the 7th term of an AP is 9. Find the arithmetic progression (AP).

Solution:

Given that, Third term of AP = 5

Seventh term of AP = 9

(i.e) a₃ = a+(3-1)d = a+2d = 5 …(1)

a₇ = a+(7-1)d = a+6d = 9 …(2)

Now, solve the equations (1) and (2), we get

a=3 and d = 1.

Therefore, the first term is 3 and the common difference is 1.

Therefore, the arithmetic progression (AP) is 3, 4, 5, 6, 7, 8, 9, ….

Question 7 : How many two-digit numbers are divisible by 3?

Solution:

The sequence of two-digit numbers which are divisible by 3 are:

12, 15, 18, 21, …, 99.

To find whether the given sequence is an AP, find the common difference.

The common difference (d) of the above-given sequence is 3, and hence, the given sequence is an Arithmetic progression (AP).

Hence, a = 12, d = 3, a_n = 99.

Now, we have to find the value of “n”.

Now, substitute the values in the formula, a_n = a+(n-1)d, we get

99 = 12+(n-1)3

99 = 12+3n-3

99-12+3 = 3n

3n = 90

n= 90/3

n=30.

Hence, there are 30 two-digit numbers that are divisible by 3.

Practice Problems of an AP

Solve the following problems:

1. Find the 30th term of an AP: 10, 7, 4, …..
2. Determine the 31st term of an AP, if its 11th term is 38 and its 16th term is 73.
3. Find the missing terms for the following AP: -4, __, __, ___, ___, 6.

FAQs on Arithmetic Sequence

Define Arithmetic Sequence.

Arithmetic Sequence is defined as the sequence where each term of the sequence can be calculated by adding a constant in the preceding term of the same sequence.

What is the Common Difference of Arithmetic Sequence?

The difference between two consecutive terms of an Arithmetic Sequence is called the common difference of an  Arithmetic Sequence.

What is the importance of the nth term formula?

The nth term formula helps:

• Find any term in an AP without listing all terms.
• Solve problems in physics, finance, and competitive exams like JEE Mains & Advanced.

Can the common difference (ddd) be negative or zero?

Yes!

• If d>0, the sequence increases.
• If d<0, the sequence decreases.
• If d=0, all terms are the same.

How is the nth term formula used in JEE and Board Exams?

It helps solve sequence-based problems quickly, such as:

• Finding missing terms in an AP.
• Checking if a number belongs to an AP.
• Word problems involving arithmetic sequences.

What is Use of Arithmetic Progression in Real-Life?

Arithmetic progression is the series that gives a common difference between two consecutive terms. It is used in daily life to generalize a set of patterns. For instance, waiting for a bus, suppose that the buses are moving at a constant speed, with the help of AP, you can tell when will the bus arrive. AP can also be used in making pyramid-like structures, etc.

Where can I download study material for the nth term of an AP?

You can download Neeraj Anand’s Arithmetic Progression PDF from ANAND CLASSES, published by ANAND TECHNICAL PUBLISHERS for Class 11, JEE, and Board exam preparation.

🚀 Master AP concepts and boost your exam scores! 💯

📖 Why Choose This Book?

✔️ Conceptual Clarity with Step-by-Step Explanations
✔️ Solved Examples & Extensive Practice Questions
✔️ Special Focus on JEE-Level Tricks & Shortcuts

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

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