Arithmetic Progression formula nth term and sum with solved examples
The concept of arithmetic progression for class 10 is essential in mathematics and helps students solve patterns and real-world problems efficiently. Mastering arithmetic progression (AP) allows class 10 students to score well in board exams and strengthens their basics for higher studies.
Understanding Arithmetic Progression for Class 10
An arithmetic progression for class 10 (often called AP for Class 10) is a sequence of numbers where the difference between any two consecutive terms remains constant. This fixed value is called the common difference (d). Arithmetic progression for class 10 is widely used in solving nth term questions, calculating the sum of AP, and identifying patterns in numbers, time, and everyday schedules.
Formula Used in Arithmetic Progression for Class 10
The standard formulas for arithmetic progression for class 10 are:
- a = first term
- d = common difference
- n = total number of terms
- l = last term
Here’s a helpful table for the core terms and formulas of arithmetic progression for class 10:
Arithmetic Progression Formula Table
| Term | Meaning | Formula/Example |
|---|---|---|
| First term (a) | Starting number of AP | E.g.: 2 in 2, 5, 8, ... |
| Common difference (d) | Constant difference | E.g.: 5−2 = 3 |
| nth term (Tn) | Any term’s value | a + (n−1)d |
| Sum of n terms (Sn) | Total of first n terms | \( S_n = \frac{n}{2}[2a + (n-1)d] \) |
This table helps visualise how arithmetic progressions for class 10 work in numbers and sequences.
Worked Example – Solving an AP Problem
Let's solve a typical board exam question on arithmetic progression for class 10 with all steps shown:
2. Identify the first term (a) and common difference (d): a = 2
d = 7 − 2 = 5
3. Use the nth term formula:
Tn = a + (n−1)d
4. Substitute n = 10:
T10 = 2 + (10−1) × 5 = 2 + 9 × 5 = 2 + 45 = 47
5. Answer: The 10th term is 47.
Practice Problems
- Find the 20th term of the AP: 3, 8, 13, ...
- What is the sum of the first 15 terms of the AP: 5, 10, 15, ...?
- Is the sequence 1, 4, 9, 16, ... an AP?
- List all arithmetic progressions for class 10 with first term 2 and common difference 4 up to 5 terms.
Common Mistakes to Avoid
- Forgetting to subtract 1 in (n−1)d when finding the nth term.
- Mistaking the sum of n terms formula with other sequence sums (like for geometric progressions).
- Mixing up 'common difference’ (d) with first term (a).
- Not checking whether the sequence fits the rules of arithmetic progression for class 10.
Real-World Applications
The knowledge of arithmetic progression for class 10 applies to daily life situations like calculating total costs, planning schedules, working out bank savings, and solving data problems involving regular intervals. Vedantu enables students to connect maths to practical problems with simple, stepwise learning.
Quick Revision Notes
- AP is a sequence with a constant difference between terms.
- nth term: a + (n−1)d
- Sum formula: \( S_n = \frac{n}{2}[2a + (n-1)d] \) or \( S_n = \frac{n}{2}(a + l) \)
- Check 'd' for every pair to confirm a sequence is AP.
- Apply formulas step by step, substituting right values.
Continue Learning – Related Topics
Explore these pages to deepen your learning on arithmetic progression for class 10 and related topics:
Arithmetic Progression
nth Term of an AP
Sequence and Series
Arithmetic Geometric Sequence
Harmonic Progression
CBSE Class 10 Maths Important Topics
Linear Equations in One Variable
Polynomials
Class 10 Maths Index
Trigonometry for Class 10
We explored the idea of arithmetic progression for class 10, how to apply AP formulas, solved examples, and saw its real-life applications. Keep practising with Vedantu to master class 10 arithmetic progression and boost your exam results!
FAQs on Arithmetic Progression for Class 10 Complete Guide
1. What is an arithmetic progression in Class 10 Maths?
An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference (d).
- If each term increases or decreases by the same number, the sequence is an AP.
- Example: 2, 5, 8, 11, ... here the common difference is 3.
- Example: 10, 7, 4, 1, ... here the common difference is -3.
2. What is the formula for the nth term of an arithmetic progression?
The formula for the nth term of an arithmetic progression is aₙ = a + (n − 1)d.
- a = first term
- d = common difference
- n = term number
- aₙ = nth term
a₅ = 3 + (5 − 1)×4 = 3 + 16 = 19.
3. How do you find the common difference in an AP?
The common difference (d) in an AP is found by subtracting any term from the next term.
- Formula: d = a₂ − a₁
- Or generally: d = aₙ − aₙ₋₁
d = 10 − 7 = 3. So, the common difference is 3.
4. How do you find the sum of first n terms of an arithmetic progression?
The sum of the first n terms of an AP is given by Sₙ = n/2 [2a + (n − 1)d].
- Sₙ = sum of first n terms
- a = first term
- d = common difference
Here, a = 2, d = 2, n = 5.
S₅ = 5/2 [2×2 + (5 − 1)×2] = 5/2 [4 + 8] = 5/2 × 12 = 30.
5. What is the formula for the sum of n terms when the last term is given?
When the last term (l) is given, the sum of n terms of an AP is Sₙ = n/2 (a + l).
- a = first term
- l = last term
- n = number of terms
S₅ = 5/2 (5 + 25) = 5/2 × 30 = 75.
6. How do you check whether a given sequence is an arithmetic progression?
A sequence is an arithmetic progression if the difference between consecutive terms is constant.
- Find the difference between first and second term.
- Find the difference between second and third term.
- If all differences are equal, it is an AP.
Differences: 4 − 1 = 3, 7 − 4 = 3, 10 − 7 = 3.
Since the difference is constant, it is an arithmetic progression.
7. How do you find the nth term from the end in an arithmetic progression?
The nth term from the end of an AP with last term l and common difference d is aₙ (from end) = l − (n − 1)d.
- l = last term
- d = common difference
- n = position from the end
14 − (2 − 1)×3 = 14 − 3 = 11.
8. What is the difference between an arithmetic progression and a geometric progression?
The main difference is that an AP has a constant difference, while a GP has a constant ratio between consecutive terms.
- In arithmetic progression (AP), each term is obtained by adding a fixed number (d).
- In geometric progression (GP), each term is obtained by multiplying by a fixed number (r).
AP: 3, 6, 9, 12 (add 3 each time)
GP: 3, 6, 12, 24 (multiply by 2 each time)
9. How do you insert arithmetic means between two numbers?
To insert arithmetic means between two numbers, first form an AP and then calculate the common difference.
- Let first term = a, last term = l.
- Total terms = number of means + 2.
- Use d = (l − a)/(n − 1).
Total terms = 5.
d = (18 − 2)/(5 − 1) = 16/4 = 4.
The AP is: 2, 6, 10, 14, 18.
The arithmetic means are 6, 10, 14.
10. What are some real-life applications of arithmetic progression?
An arithmetic progression is used in real life wherever quantities increase or decrease by a constant amount.
- Calculating monthly savings with fixed deposits.
- Finding total salary with fixed annual increments.
- Arranging seats in rows with equal increase per row.
- Calculating simple interest over equal intervals.
