Class 10 chapter 5 arithmetic progression

1. Sequence: A set of numbers arranged in some definite order and formed according to some rules is called a sequence.
2. Progression: The sequence that follows a certain pattern is called progression.
3. Arithmetic Progression: A sequence in which the difference obtained by subtracting any term from its preceding term is constant throughout, is called an arithmetic sequence or arithmetic progression (A.P.).
   The general form of an A.P. is a, a+d, a +2d, ….. (a : first term, d = common difference).
   The terms of A.P. is denoted by a₁, a₂, a₃,……..,a_n.
4. General Term: If ‘a’ is the first term and ‘d’ is common difference in an A.P., then n^th term (general term) is given by a_n = a + (n − 1) d

Sum of n Terms of an A.P.: If ‘a’ is the first term and ‘d’ is the common difference of an A.P., then sum of first n terms is given by
S_n = ⁿ/₂{2a + (n − 1) d}

If ‘l’ is the last term of a finite A.P. then the sum is given by S_n = ⁿ/₂ {a +l}
(i) If an is given, then common difference d = a_n-a_n-1.
(ii) If Sn is given, then n^th term is given by a_n=S_n-S_n-1
(iii) If a, b, c are in A.P., then 2b = a + c.
(iv) If a sequence has n terms, its r^th term from the end = (n-r+1)^th term from the beginning.