Sequence & Series – AP nth Term | Complete Arithmetic Progression Concept for JEE Main

Sequence & Series – AP nth Term

Arithmetic Progression, commonly called AP, is one of the most important basic concepts of Sequence & Series. JEE Main questions from AP can look very simple, but small mistakes in the term number or sign of the common difference can easily lead to a wrong answer.

The most important formula of this topic is:

a_n = a + (n − 1)d

Once you understand this formula properly, you can solve questions involving the nth term, finding which term is given, finding the first term and common difference from two terms, terms from the end, and many other AP-based questions.

---

1. What is an Arithmetic Progression?

An Arithmetic Progression (AP) is a sequence in which the difference between every two consecutive terms remains constant.

For example:

3, 7, 11, 15, 19, ...

Check the differences:

• 7 − 3 = 4
• 11 − 7 = 4
• 15 − 11 = 4
• 19 − 15 = 4

Since the difference is always 4, this sequence is an AP.

The general form of an AP is:

a, a + d, a + 2d, a + 3d, ...

Here, d is called the common difference.

It is calculated as:

d = a₂ − a₁

---

2. Important Terms in AP

Term	Meaning
a	First term of the AP
d	Common difference
an	nth term of the AP
n	Term number
l	Last term, when the AP is finite

For an AP:

a, a+d, a+2d, a+3d, ...

the first term is a and the common difference is d.

---

3. Deriving the nth Term Formula ⭐

Let us write the first few terms of an AP:

Term	Value
1st term	a
2nd term	a + d
3rd term	a + 2d
4th term	a + 3d
nth term	a + (n − 1)d

Notice the pattern:

• 1st term → 0d
• 2nd term → 1d
• 3rd term → 2d
• 4th term → 3d

Therefore, for the nth term, the multiplier of d is always n − 1.

⭐ a_n = a + (n − 1)d

JEE Shortcut

Whenever the question asks for the nth term, immediately think:

Term number n → (n − 1)d

This one observation prevents one of the most common AP mistakes.

---

4. Example: Find the 20th Term

Find the 20th term of:

3, 7, 11, 15, ...

First identify:

• a = 3
• d = 7 − 3 = 4
• n = 20

Using:

a_n = a + (n − 1)d

We get:

a₂₀ = 3 + (20 − 1)(4)

a₂₀ = 3 + 19 × 4

a₂₀ = 3 + 76

∴ a₂₀ = 79

---

5. Finding Which Term is Given

Sometimes JEE Main gives a particular value and asks:

“Which term of the AP is equal to this value?”

In such questions, put the given value equal to the nth-term formula and solve for n.

Example

Which term of the AP:

5, 9, 13, 17, ...

is equal to 81?

Here:

• a = 5
• d = 4
• a_n = 81

Therefore:

81 = 5 + (n − 1)4

76 = 4(n − 1)

19 = n − 1

n = 20

Therefore, 81 is the 20th term of the AP.

---

6. Increasing, Decreasing and Constant AP

The sign of the common difference tells us how the AP behaves.

Condition	Type	Example
d > 0	Increasing AP	2, 5, 8, 11, ...
d < 0	Decreasing AP	20, 17, 14, 11, ...
d = 0	Constant AP	7, 7, 7, 7, ...

Important: The common difference does not have to be positive. A negative value of d is completely valid.

---

7. nth Term from the Last ⭐

Another useful AP concept is finding a term when counting from the end of the AP.

Suppose the last term is l.

The terms from the end are:

l, l − d, l − 2d, l − 3d, ...

Therefore:

kth term from the end = l − (k − 1)d

Example

Consider the AP:

3, 7, 11, ..., 99

Find the 5th term from the end.

Here:

• l = 99
• d = 4
• k = 5

Therefore:

T₅ = 99 − (5 − 1)(4)

T₅ = 99 − 16

T₅ = 83

---

8. Finding AP from Two Given Terms 🔥

JEE Main frequently gives two terms of an AP and asks you to find the first term, common difference or another term.

Suppose a_m and a_n are known.

We know:

a_m = a + (m − 1)d

a_n = a + (n − 1)d

Subtracting the two equations:

a_n − a_m = (n − m)d

Therefore:

d = (a_n − a_m) / (n − m)

Once d is known:

a = a_n − (n − 1)d

Example

Given:

a₅ = 17

a₁₀ = 32

Find the first term and common difference.

Using:

d = (32 − 17)/(10 − 5)

d = 15/5 = 3

Now:

a₅ = a + 4d

17 = a + 4(3)

a = 5

Therefore the AP is:

5, 8, 11, 14, 17, ...

---

9. Three Numbers in AP

When three numbers are in AP, they can be conveniently represented as:

a − d, a, a + d

Here, the middle term is exactly the average of the first and third terms.

Therefore:

2 × Middle = First + Third

Example

Suppose:

x, 10, 16

are in AP.

Using:

2 × Middle = First + Third

2(10) = x + 16

20 = x + 16

x = 4

Hence the three terms are:

4, 10, 16

---

10. Special APs

Some APs appear repeatedly in mathematical problems and are worth remembering.

AP	nth Term
Natural numbers: 1, 2, 3, 4, ...	an = n
Even numbers: 2, 4, 6, 8, ...	an = 2n
Odd numbers: 1, 3, 5, 7, ...	an = 2n − 1
Multiples of k: k, 2k, 3k, ...	an = nk

These formulas can save time in JEE Main when the sequence is a standard number pattern.

---

11. AP nth Term as a Linear Function

The nth-term formula is:

a_n = a + (n − 1)d

Expand it:

a_n = a + nd − d

Therefore:

a_n = dn + (a − d)

This has the same form as a straight-line equation:

y = mx + c

Therefore, if we consider a_n as a function of n:

• Slope = d
• Intercept = a − d

So, a_n versus n represents a straight line.

This is a useful connection between Sequence & Series and the concept of a linear function.

---

12. JEE Main-Level Solved Question 🔥

Question: The 7th term of an AP is 20 and the 15th term is 52. Find the 25th term.

Step 1: Write the two equations

For the 7th term:

a + 6d = 20

For the 15th term:

a + 14d = 52

Step 2: Subtract

(a + 14d) − (a + 6d) = 52 − 20

8d = 32

d = 4

Step 3: Find the first term

Using:

a + 6d = 20

a + 6(4) = 20

a = −4

Step 4: Find the 25th term

a₂₅ = a + 24d

a₂₅ = −4 + 24(4)

a₂₅ = −4 + 96

✅ a₂₅ = 92

---

13. Faster JEE Method

In many questions, you do not even need to find the first term.

From:

a₇ = 20

and

a₁₅ = 52

we already know:

a₁₅ − a₇ = (15 − 7)d

So:

52 − 20 = 8d

d = 4

Now:

a₂₅ = a₁₅ + (25 − 15)d

a₂₅ = 52 + 10(4)

a₂₅ = 92

JEE Trick: If one term is already known, move directly from that term to the required term using the number of steps multiplied by d.

---

14. Common JEE Traps ⚠️

Trap 1: Forgetting −1

Wrong:

a_n = a + nd

Correct:

a_n = a + (n − 1)d

Trap 2: Wrong 3rd Term

Wrong:

3rd term = a + 3d

Correct:

3rd term = a + 2d

Trap 3: nth Term from End

Wrong:

l − kd

Correct:

l − (k − 1)d

Trap 4: Assuming d is Always Positive

This is incorrect.

d can be positive, negative or zero.

• d > 0 → Increasing
• d < 0 → Decreasing
• d = 0 → Constant

Trap 5: Confusing Term Number with Number of Steps

To reach the 10th term from the first term, you make 9 steps, not 10.

That is exactly why the formula contains:

(n − 1)d

---

15. JEE Main Quick-Solving Strategy

1. Identify the first term a.
2. Find the common difference d.
3. Identify the required term number n.
4. Use a_n = a + (n − 1)d.
5. If two terms are given, first find d.
6. If counting from the end, use l − (k − 1)d.
7. For three terms in AP, use 2 × Middle = First + Third.
8. Always check the sign of d.

Most basic AP questions can be solved in a few lines if these steps become automatic.

---

16. AP Formula Sheet 📌

Concept	Formula
General AP	a, a+d, a+2d, ...
Common difference	d = a2 − a1
nth term	an = a + (n − 1)d
kth term from end	l − (k − 1)d
Common difference from two terms	d = (an − am)/(n − m)
Three terms in AP	a − d, a, a + d
Three-term condition	2 × Middle = First + Third
Increasing AP	d > 0
Decreasing AP	d < 0
Constant AP	d = 0

---

17. One-Minute Revision 🚀

Before entering the JEE exam, remember this simple chain:

AP → Constant Difference → d

First term → a

nth term → a + (n − 1)d

From end → l − (k − 1)d

Two terms → Find d first

Three terms → a − d, a, a + d

Three terms → 2 × Middle = First + Third

d > 0 → Increasing

d < 0 → Decreasing

d = 0 → Constant

---

18. Final Revision Box 🔥

Arithmetic Progression: a, a+d, a+2d, ...

Common Difference: d = a₂ − a₁

nth Term: a_n = a + (n − 1)d

kth Term from End: l − (k − 1)d

Two Given Terms: d = (a_n − a_m)/(n − m)

Three Terms in AP: a − d, a, a + d

Three-Term Condition: 2 × Middle = First + Third

d > 0: Increasing

d < 0: Decreasing

d = 0: Constant

---

19. Practice Questions for JEE Main

1. Find the 25th term of the AP 7, 12, 17, 22, ...
2. Which term of the AP 4, 9, 14, 19, ... is equal to 99?
3. If a₆ = 22 and a₁₂ = 46, find the common difference.
4. If the 8th term of an AP is 31 and the 18th term is 71, find the 28th term.
5. Find x if x, 15 and 23 are in AP.
6. Find the 7th term from the end of the AP 2, 6, 10, ..., 98.

Tip: Before solving, first identify a, d, n. This simple habit will make AP questions much faster.

---

20. PDF Notes

You can also add your detailed PDF notes below for students who want to revise the topic offline.

---

21. Frequently Asked Questions (FAQs)

Q1. What is the nth term of an AP?

The nth term of an AP is:

a_n = a + (n − 1)d

Q2. Why is there an (n − 1) in the AP formula?

Because the first term already contains a. To reach the second term, one common difference is added; to reach the third term, two common differences are added. Therefore, the nth term requires n − 1 differences.

Q3. Can the common difference be negative?

Yes. If d < 0, the AP is decreasing.

Q4. How do I find the common difference when two terms are given?

Use:

d = (a_n − a_m)/(n − m)

Q5. How are three numbers represented when they are in AP?

Three numbers in AP can be represented as:

a − d, a, a + d

Q6. What is the condition for three numbers to be in AP?

The middle number must be the average of the other two:

2 × Middle = First + Third

Q7. What is the kth term from the end of an AP?

If the last term is l, then the kth term from the end is:

l − (k − 1)d

Q8. What is the nth term of the even-number sequence?

For 2, 4, 6, 8, ..., the nth term is:

a_n = 2n

---

Final Thoughts

Arithmetic Progression is a highly manageable topic once the pattern of terms becomes clear. The central idea is simple: the difference between consecutive terms remains constant.

For JEE Main, the most important formula is:

a_n = a + (n − 1)d

But don't just memorize it. Understand why the formula contains (n − 1). Also remember the formulas for terms from the end, finding d from two given terms, and representing three terms in AP.

If these patterns become automatic, many AP questions can be solved quickly and accurately during the JEE Main exam.

Master the pattern, remember the formula, avoid the traps — and AP becomes one of the easiest scoring areas in Sequence & Series.