PROGRESSIONS AND SERIES
There are three progressions or series namely,
- Arithmetical Progression
- Geometrical Progression
- Harmonical These series are discussed below.
ARITHMETICAL PROGRESSION
(1) An arithmetical progression is represented as a, a+d, a+2d, a+3d, a+(n-1)d. Where a is the first term and d is the common difference. The difference between 2nd and first tern, third and 2nd term and so on is same throughout the series, e.g. if a, b, c, d are in A.P then b-a = c-d = d-c.
2) The nth term of an A.P = a + (n-l)d where a is the first term, n the total number of terms and d is the common difference.
- The sum of n terms of an AP, a, + (a+d) + a + ..is n
Sn = — [sa + (n – l)d] 2
- Arithmetic Mean (A.M)
a + b between a and b is given by A.M = ———–
2
- n arithmatic means between a and b are given by b – a 2(b – a) n(b – a)
a + …+
n + 1, n + 1 n + 1
Some Result of an A.P.
- a) In solving the problems of an P., in which the sum of three consecutive terms is given, then the three terms should be taken as a-d, a, a+d
(b) For solving the problems of four terms where the sum of four terms is given then the four terms are taken as a-3d, a-d, a+d and a+3d.
GEOMETRICAL PROGRESSION
- In a geometrical progression the ratio of the 2nd and first term, the 3rd term and second term and so on is the This is called common ratio.
A geometrical progression is represented as a, ar, ar2, ar2, ar3…arn-1 where a is the first term and r is the common ration.
- The nth term of P., a, ar, ar2, ar3…….. s given as = arn-1
- The sum of n terms of a,G.P., a, ar, ar2.. is given by
|
a(l – rn) a(r – 1)
Sn = ———- when r < 1 or Sn =—————————————————– whern r > 1
l – r l – r
a
- Sum of a P. up to infinity is given by Sn = when r < 1
l – r
- The geometrical mean between a anb b in a P is given by G = Öab
- Between a and b we can inert n geometric means by using the following
formulae
b a —
a
1
——-
n + 1
b
, a —
a
2
——-
n + 1
b a —
a
n
——
n + 1
b m
or Gm = a — ——-
a n + 1
- If we multiply n geometric means between a and b, then the result is (Öab)n
HARMONICAL PROGRESSION
- If we represent a harmonic Progression as a, b, c, d,. then
1 1 1 1
— , —, —, —,…. are in A.P.
- b c d
In this case, the differences 1 1 1 1 1 1
— – — = — – — = — – — =…..
- a c b d c
- The harmonic mean between a and b is given by 1 1 1 1 2ab
= + or H =
H 2 a b a + b
- If nth term is = an2+b n+c then the sum sn = a + ån2+b å n + cn
- If we have non-zero numbers as a, a2,. an then l+.M of these numbers is
1 1 1 1 1
= + +…+
H n a1 a2 an
sum of Natural numbers, their squares, cubes etc.
n(n + 1) (a) 1+2+3+…+n = ———–
2 (b) 12 + 22 + 32 + … + n2
n(n + 1)(2n + 1)
=
6
n(n +1) 2
(c) 13 + 23 + 33 + …. + n3 = ———–
2
(d) 2 + 4 + 6 +… to n terms = n2
(e) 2 + 4 + 6 +… to n terms = n(n + 1)
Arithmetic-Geometric sequence : If a, a + d, a + 2d… is in A.P. and a, ar, ar2…is in G.P. then a sequence of the form a, (a + d)r, (a + 2d) r2 +………………………………………………………… is known as
arithmetic-geometric series. Its nth term tn = {a + (x – 1)d} ln-1 and sum of this series is
1 d.r
sa =
l – r
+
(l – r)2
