PROGRESSION SERIES
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) 1+23+33+43+…..+n3=
{1/2 n(n+1)}2
Harmonic Progression Formula:
The general form of a harmonic progression:
1/a, 1/(a+b), ........, 1/(a+(n-1)d)
The nth term of a Harmonic series is:
Tn= 1/(a+(n-1)d)
In order to solve a problem on Harmonic Progression, one should make the corresponding AP series and then solve the problem.
Harmonic Mean: If three terms a, b, c are in HP, then 1/a, 1/b and 1/c form an A.P.
Therefore, harmonic mean formula-
2/b = 1/a + 1/c
The harmonic mean b = 2ac /(a + c)
Some Special Formulas:
Sum of the cubes of first n natural numbers = {
Sum of first n natural numbers = [n(n+1)] / 2
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