ARITHMETIC PROGRESSION

Class 10 Maths Formulas

CHAPTER 5 ARITHMETIC PROGRESSION
1.	A sequence a₁, a₂, … , aₙ is an Arithmetic Progression (A.P) if the difference between consecutive terms is constant, called the common difference d (positive or negative).
2.	General A.P is: a, a + d, a + 2d, a + 3d, … where a is the first term and d is the common difference.
3.	The n^th term of an A.P is: aₙ = a + (n − 1)d (a₁₀ = a + 9d ; a₁₆ = a + 15d)
4.	Easy method to find a term from the end of an A.P: Find the 7^th term from the end of the sequence: 17, 14, 11, … , −40 Write from the end: −40, −37, … , 17 For this A.P: a = −40, d = 3 a₇ = a + 6d = −40 + (6 × 3) = −22
5.	To check whether a number belongs to an A.P: Find the first term a and common difference d. Use the formula: aₙ = a + (n − 1)d Substitute the given number and find n. If n is a positive whole number, the number belongs to the A.P.
6.	Sum of n terms of an A.P is denoted by Sₙ. Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l) where l is the last term.
7.	Sum of first n natural numbers: 1 + 2 + 3 + … + n = n(n + 1)/2
8.	Sum of first n odd natural numbers: 1 + 3 + 5 + … + (2n − 1) = n²
9.	Sum of first n even natural numbers: 2 + 4 + 6 + … + 2n = n(n + 1)
10.	S₁ = a₁ (first term) S₂ = a₁ + a₂ S₃ = a₁ + a₂ + a₃ S₂ − S₁ = a₂ S₃ − S₂ = a₃ In general: aₙ = Sₙ − Sₙ₋₁