Aptitude - Progression

Sequence

A succession of numbers formed and arranged in a definite order according to a certain definite rule is called a sequence.

Arithmetic Progression (A.P.)

It is a sequence in which each term, except the first one differs the preceding term by a constant. This constant is called the common difference. We denote the first term by a, common difference by d, nth term by Tṇ and the sum of first n terms by Sṇ.

Examples

5, 8,11,14,17...is an A.P. in which a=5 and d = (8-5) =3.
8, 5, 2,-1,-4,-7.... is an A.P. in which a = 8 and d = (5-8) = -3.

General Term of an A.P.

In a given A.P., let first term =a, common difference =d. Then,

Tn= a + (n-1) d.
Sum of n terms of an A.P.
Sn = n/2[2a+ (n-1) d]
Sn = n/2 (a + L), where L is the last term.

Geometrical Progression (G.P.)

A sequence in which each term, except the first one bears a constant ratio with its preceding term, is called a geometrical progression, written as G.P. The constant ratio is called the common ratio of the G.P. We denote its first term by a and common ratio by r.

Example

2, 6, 18, 54, is a G.P.in which a=2 and r=6/2=3.
24, 12, 6, 3... Is a G.P. in which a = 24 and r = 12/24=1/2.

General Term of a G.P.: In a G.P. we have

Tn= arn-1
Sum of n terms of a G.P.
Sn = a (1-rn)/ (1-r), When r < 1
a (r - 1n)/(r-1), When r > 1

Arithmetic Mean

A.M. of a and b = 1/2(a+b).

Geometric Mean

G.M. of a and b =√ab

Some General Series

(i) 1+2+3+4+…….+n=1/2n (n+1).
(ii) 12+22+32+42+……+n2 = n(n+1)(2n+1)/6 
(iii)  13+23+33+43+…..+n3= {1/2 n(n+1)}2

Solved Examples