Overview
This lecture fully covers all the main topics of the Class 10 Mathematics chapter "Arithmetic Progression (AP)", including their definitions, formulas, and important solved problems.
Definition of Arithmetic Progression
- Arithmetic Progression (AP) is a sequence in which a fixed number (common difference) is added to each subsequent term.
- AP is generally written as a, a+d, a+2d, a+3d... where a is the first term and d is the common difference.
- Each term = previous term + common difference.
Vocabulary and Basics
- First term (a), next terms (a2, a3тАж), formula for the nth term: an = a + (n-1)d
- Common difference (d) = next term - previous term (a2-a1, a3-a2 ...).
- To check whether a sequence is AP or not, the difference between consecutive terms must be the same.
Important Examples
- d can be positive, negative, or zero.
- If d is constant, it is AP; otherwise, it is not AP.
- For example: 2, 4, 6, 8тАж (d=2), -1.2, -3.2, -5.2тАж (d=-2)
Methods to Solve AP Problems
- Formula for the nth term: an = a + (n-1)d
- Formula to find sum (Sum): Sn = n/2 [2a + (n-1)d] or Sn = n/2 (a + l), where l = last term
- Solve mixed problems involving m, n, p, q, etc. by forming two simultaneous equations.
Special Types of Problems
- Which term has the value ___ / which term is zero: put the desired value in an and solve for n.
- How many three-digit numbers divisible by 7 are in AP: find the first and last such numbers in AP and then find n.
- To find the middle term:
- For odd n, the middle term is the (n+1)/2th term.
- For even n, both the n/2th and (n/2 +1)th terms are middle terms.
Key Terms & Definitions
- Arithmetic Progression (AP) тАФ A sequence of numbers where a fixed number is added to each subsequent term.
- First term (a) тАФ The first value of the AP.
- Common difference (d) тАФ The difference between any two consecutive terms.
- nth term (an) тАФ The nth term of the AP, formula: a + (n-1)d.
- Sum of AP (Sn) тАФ The sum of the first n terms, formula: n/2 [2a + (n-1)d] or n/2(a+l)
Action Items / Next Steps
- Solve all exercise questions of the NCERT AP chapter.
- Note down the main formulas in your copy and revise them.
- Practice AP questions from previous years' question papers.
- Prepare for the next class on Polynomials.