Sequence and Series – JEE Mains Mathematics

1. Arithmetic Progression (A.P.)

A sequence where the difference between consecutive terms is constant.
General form: a, a+d, a+2d, ..., a+(n−1)d
n-th term: T_n = a + (n−1)d
Sum of first n terms: S_n = (n/2)[2a + (n−1)d] or S_n = (n/2)(a + l)

2. Geometric Progression (G.P.)

A sequence where each term is obtained by multiplying the previous term by a constant ratio.
General form: a, ar, ar², ar³, ..., arⁿ⁻¹
n-th term: T_n = arⁿ⁻¹
Sum of first n terms (r ≠ 1): S_n = a(1 − rⁿ)/(1 − r)
Sum to infinity (|r| < 1): S = a/(1 − r)

3. Insertion of A.M. and G.M. Between Two Numbers

To insert n arithmetic means between a and b:
Use A.P. formula with a as the first term and b as the (n+2)^th term.
d = (b − a)/(n+1), then insert values using a + d, a + 2d, ...
To insert n geometric means between a and b:
Use G.P. with a as the first term and b as the (n+2)^th term: b = arⁿ⁺¹
r = (b/a)^1/(n+1)

4. Relation Between A.M. and G.M.

For any two positive numbers a and b:
A.M. = (a + b)/2
G.M. = √(ab)
Relation: A.M. ≥ G.M., with equality only when a = b