Those sequences whose terms follow certain pattern are called progression.

We define three types of progression

Arithmetic progression (A.P.) : If a is first term and d is common difference then A.P. can be written as

a, a + d, a + 2d, .......a + (n – 1)d, .......then

t n = a + (n – 1)d where n = number of terms in A.P.

S n = [2a + (n – 1)d] t n = n th term of A.P.

S n = sum of n term of A.P.

Geometric Progression (G.P.) : If a is first term and r is common ratio then G.P. can be written as

a, ar, ar 2 ,......... ar n–1 .........

n th term of GP = t n = a r n − 1

Sum of the first n terms of GP

S n =

Sum of an infinite terms of GP when ⏐ r ⏐ < 1. When n → ∞, r n → 0 if ⏐ r ⏐ < 1 therefore,

S ∞ = .

Harmonic progression (H.P.) : A sequence of non-zero number is said to be in H.P if the reciprocals of its terms are in A.P.. If the sequence a 1 , a 2 , a 3 ,...., a n is in H.P. then 1/a 1 , 1/a 2 ,...., 1/a n is in A.P.

(i) If 9, 12, 15, 18, ...... is any progression then