How to Find the nth Term and Sum of Arithmetic Progression in Class 10?
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 – Concept, Formulas & Solutions
1. What is arithmetic progression for class 10?
An Arithmetic Progression (AP) for Class 10 is a sequence of numbers where the difference between consecutive terms is constant. It is used to solve various board-level math problems involving sequences and series, helping students understand patterns and sums effectively.
2. What are the key formulas for AP in class 10?
The key formulas for Arithmetic Progression in Class 10 include:
• The nth term formula: Tn = a + (n - 1)d, where a is the first term, d is the common difference, and n is the term number.
• The sum of n terms formula: Sn = n/2 [2a + (n − 1)d] or Sn = n/2 (a + l), where l is the last term.
These formulas are essential for solving problems related to AP sequences and series.
3. Is 1, 2, 3, 4, 5, 6 an arithmetic progression?
Yes, the sequence 1, 2, 3, 4, 5, 6 is an Arithmetic Progression (AP) because the difference between consecutive terms is constant (difference of 1). This fits the definition of AP where d = 1.
4. How do I find the nth term of an AP?
To find the nth term of an AP, use the formula Tn = a + (n − 1)d, where:
• a = first term
• d = common difference
• n = position of the term
Substitute these values into the formula and simplify to get the nth term.
5. How can I practice AP questions for board exams?
Practice AP questions by solving:
• Step-by-step example problems covering nth term and sum of terms.
• Worksheets and sample papers focused on Class 10 boards.
• Real-life application problems to develop problem-solving skills.
Use revision notes and formula boxes to reinforce concepts effectively.
6. Where can I download worksheets and PDFs for AP class 10?
You can download Arithmetic Progression worksheets and PDFs for Class 10 from trusted educational websites like Vedantu, NCERT, and CBSE portals. These resources provide notes, solved examples, and practice sets aligned with the Class 10 syllabus for comprehensive preparation.
7. Why do students confuse arithmetic progression with geometric progression?
Students often confuse Arithmetic Progression (AP) with Geometric Progression (GP) because both involve sequences. The key difference is:
• In AP, the difference between terms is constant.
• In GP, the ratio between consecutive terms is constant.
Understanding this fundamental difference helps avoid confusion in problems.
8. What common mistakes cause wrong AP answers in exams?
Common mistakes in solving AP problems include:
• Incorrect calculation of the common difference (d).
• Using wrong formulas or mixing up nth term and sum of terms formulas.
• Not substituting accurate values while finding terms.
• Misunderstanding finite versus infinite AP.
Careful reading and stepwise solving minimize such errors.
9. Why isn’t the formula for AP sum used for all AP types?
The sum of n terms formula is valid only for finite APs because it requires knowing the number of terms. For infinite APs, the sum may not converge unless specific conditions are met. Hence, sum formulas are applied based on the nature of the sequence.
10. How is AP used in everyday life and data problems?
Arithmetic Progression finds applications in everyday life such as:
• Calculating monthly savings or salaries with fixed increments.
• Scheduling repeated events at regular intervals.
• Analyzing patterns in data sequences.
Understanding AP helps in solving practical problems involving consistent increases or decreases.
11. Why do calculators sometimes give AP sum errors?
Calculators may give errors in AP sum calculations due to:
• Incorrect input of values for a, d, or n.
• Using formulas meant for infinite series on finite sequences or vice versa.
• Precision issues with large numbers.
Careful formula application and verification prevent such errors.
12. What is the difference between finite and infinite AP?
A finite AP has a limited number of terms, with a definite last term, while an infinite AP continues indefinitely without an end. Sum formulas apply differently:
• Finite AP sums are calculated using standard sum formulas.
• Infinite AP sums require convergence criteria and may not always be calculable.
