Arithmetic Progressions Class 10
Download the PDF containing Arithmetic Progressions Class 10 notes and a mind map for chapter revision. Understand the common difference, nth term and sum formulas through clear explanations, worked examples and practice questions.
A seating arrangement with three additional chairs in each row, or a savings plan that increases by a fixed amount each month, follows a predictable numerical pattern. Class 10 Arithmetic Progression helps you describe such patterns and calculate a particular term or the total of several terms without listing them all.
These Arithmetic Progression Class 10 notes explain how to recognise an AP, choose the correct formula and interpret an answer. Use the detailed notes for understanding and the Arithmetic Progressions Class 10 mind map for quick recall. Then practise questions independently to check whether you can apply the ideas in different situations.
What Is an Arithmetic Progression?
An arithmetic progression, or AP, is a sequence in which the difference between each term and the preceding term remains constant. This fixed difference is called the common difference, represented by d.
Recognising an AP
Consider 4, 9, 14, 19, …
9 − 4 = 5
14 − 9 = 5
19 − 14 = 5
The differences are equal, so the sequence is an AP with
first term a = 4 and common difference d = 5.
In contrast, 2, 4, 8, 16, … is not an AP because its consecutive differences are 2, 4 and 8. Always subtract consecutive terms in the same order when checking a sequence.
Arithmetic Progression Class 10 All Formulas
In this formula sheet, a is the first term, d is the common difference, n is a positive integer representing a position or number of terms, l is the last term of a finite AP, and Sn is the sum of the first n terms.
| Quantity | Formula | When to Use It |
|---|---|---|
| General form | a, a + d, a + 2d, a + 3d, … | To describe consecutive terms. |
| Common difference | d = ak+1 − ak | Subtract any term from the next term. |
| nth term | an = a + (n − 1)d | To find a single term or its position. |
| Sum of the first n terms | Sn = (n/2)[2a + (n − 1)d] | When a, d and n are known. |
| Sum using the last term | Sn = (n/2)(a + l) | When a, l and n are known. |
| Number of terms | n = (l − a)/d + 1 | d ≠ 0; the result must be a positive integer. |
| rth term from the end | l − (r − 1)d | For a finite AP with at least r terms. |
| Term from consecutive sums | an = Sn − Sn−1 | Use S0 = 0. |
| Arithmetic mean between x and y | (x + y)/2 | To form three terms x, A, y in AP. |
Do Not Confuse a Term with a Sum
an is the value of one term, while Sn is the total of the first n terms. For 2, 4, 6, 8, …, the fourth term is 8, but the sum of the first four terms is 20.
Why Does the nth-Term Formula Use n − 1?
The first term is already a. Reaching the second term requires one addition of d; reaching the third requires two. Therefore, reaching the nth term requires n − 1 additions of the common difference.
a1 = a
a2 = a + d
a3 = a + 2d
Therefore, an = a + (n − 1)d.
Class 10 Arithmetic Progression Questions with Solutions
Example 1: Find the 15th Term
Question: Find the 15th term of 6, 10, 14, 18, …
a = 6, d = 4, n = 15
a15 = 6 + (15 − 1) × 4
a15 = 6 + 56
a15 = 62
Example 2: Which Term Is 94?
Question: Which term of 4, 9, 14, 19, … is 94?
a = 4, d = 5
94 = 4 + (n − 1) × 5
90 = 5(n − 1)
18 = n − 1
n = 19
Since n is a positive integer, 94 is the 19th term. A fractional or non-positive position would not identify a term in this AP.
Sum of the First n Terms of an AP
To understand the sum formula, write a finite AP forwards and backwards. Each paired column totals a + l, and there are n columns. Thus 2Sn = n(a + l), giving Sn = (n/2)(a + l). Substituting l = a + (n − 1)d produces the other sum formula.
Example 3: Find the Sum of the First 20 Terms
Question: Find the sum of the first 20 terms of 3, 7, 11, 15, …
a = 3, d = 4, n = 20
S20 = (20/2)[2 × 3 + (20 − 1) × 4]
S20 = 10[6 + 76]
S20 = 10 × 82
S20 = 820
Example 4: Find the First Term and Common Difference
Question: The fourth term of an AP is 17 and the ninth term is 37. Find a and d.
a + 3d = 17
a + 8d = 37
Subtracting the first equation from the second:
5d = 20
d = 4
a + 12 = 17
a = 5
Arithmetic Progression Class 10 Application Questions
Example 5: Seats in an Auditorium
Question: An auditorium has 12 rows. The first row has 18 seats, and each following row has 2 more seats than the previous row. Find the total number of seats.
a = 18, d = 2, n = 12
S12 = (12/2)[2 × 18 + (12 − 1) × 2]
S12 = 6[36 + 22]
S12 = 6 × 58
Total seats = 348
The question asks for all seats, so use the sum formula. The nth-term formula alone would give only the seats in the last row.
Arithmetic Progression Class 10 HOTS
Example 6: Find the First Negative Term
Question: Find the first negative term of 25, 21, 17, 13, …
a = 25, d = −4
an = 25 + (n − 1)(−4)
an = 29 − 4n
For a negative term, 29 − 4n < 0.
n > 29/4
n > 7.25
The smallest possible integer is n = 8.
a8 = 29 − 32
First negative term = −3, at position 8.
Arithmetic Progressions Class 10 Mind Map
Use the mind map of Arithmetic Progression Class 10 to connect the definition, common difference, nth term and sum. Ask yourself whether a question requires a single value, a position or a total before choosing the formula.
For active revision, cover the formula section and write the main results from memory. Explain what each symbol means, then solve one question from each category. Return to the detailed notes when you cannot explain why a formula applies.
Important Questions for Quick Practice
-
Find the common difference of 15, 11, 7, 3, …
Answer: −4. -
Find the 10th term of 7, 10, 13, …
Answer: 34. -
Find the sum of the first 15 positive even integers.
Answer: 240. -
Is 50 a term of 3, 7, 11, …?
Answer: No. Solving gives n = 12.75, which is not an integer. -
Find the arithmetic mean between 8 and 20.
Answer: 14.
NCERT Exercises and Exam Revision
Use these notes alongside Arithmetic Progression Class 10 Exercise 5.1, 5.2 and 5.3 in the applicable NCERT edition. Match the actual question with your textbook because exercise numbering can differ between boards and editions.
Build confidence through routine questions before attempting HOTS, application problems and verified previous year questions with solutions. Use an Arithmetic Progression Class 10 test paper as independent practice, then check formula selection, substitution and interpretation. Avoid rounding the number of terms: a valid term count must be an integer.
Related Notes and Worksheet Practice
Apply the formulas using the Arithmetic Progression Class 10 Worksheet with Solutions . Attempt the questions before checking the solutions.
Revise equation-solving skills with Quadratic Equations Class 10 Notes and Mind Map , or explore the CBSE Class 10 Maths Notes and Mind Maps collection .
Frequently Asked Questions
Does this PDF contain Arithmetic Progressions notes and a mind map?
Yes. The PDF combines notes for understanding the chapter with a mind map for visual revision. Use the explanations first, then recall the formulas and connections through the map.
How do I check whether a sequence is an AP?
Subtract each term from the following term. If all consecutive differences are equal, the sequence is an AP. For example, 2, 5, 8, 11 has a constant difference of 3.
Can the common difference be negative or zero?
Yes. A decreasing AP has a negative common difference, such as 10, 7, 4, 1 with d = −3. A constant sequence such as 5, 5, 5, 5 is also an AP, with d = 0.
What is the difference between aₙ and Sₙ?
an is one term at position n. Sn is the sum of the first n terms. In 3, 6, 9, …, a3 = 9, while S3 = 18.
How do I check whether a number belongs to an AP?
Set the number equal to a + (n − 1)d and solve for n. It belongs to the AP when n is a positive integer within any stated finite range. If d = 0, every term equals a.
Which sum formula should I use?
Use Sn = (n/2)[2a + (n − 1)d] when the common difference is known. Use Sn = (n/2)(a + l) when the first term, last term and number of terms are known.
How do I find a term from the end?
For a finite AP, use l − (r − 1)d for the rth term from the end. For example, in 2, 5, 8, 11, 14, the third term from the end is 14 − 2 × 3 = 8.
How can I find the nth term when the sum is given?
Use an = Sn − Sn−1. Subtracting the sum of the first n − 1 terms leaves only the nth term. For the first term, use a1 = S1.
What if a sum question gives two possible values of n?
Check both values against the requirement that n is a positive integer and against the problem’s conditions. A decreasing AP can sometimes reach the same total at two different valid term counts, so do not automatically reject the larger value.
How do I represent three numbers in AP?
A useful form is a − d, a, a + d. Their sum is 3a, which often simplifies questions involving a given sum. The common difference between consecutive terms is d.
What are common mistakes in Class 10 AP questions?
Using a + nd instead of a + (n − 1)d, reversing subtraction when finding d, confusing a term with a sum and accepting fractional term positions are common mistakes. Record the given values and the required quantity before substituting.
Where can I practise Arithmetic Progression questions with solutions?
Visit the Arithmetic Progression Class 10 worksheet with solutions . Use it after revising the notes and mind map to practise term, sum and application questions independently.
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