Sequence
A
sequence is a function defined on set N = {1, 2, 3, 4, 5, …} of the positive
integers. Since, the domain of every sequence is the set N of natural numbers,
so the sequence is represented by its range.
The image
of 1, 2, 3, …, n, … under the function f are generally denoted by t1,
t2, t3…, i.e., t1
= f(1), t2 = f(2), t3 = f(3), …, tn = f(n) …..
Series
If t1,
t2, t3, … tn … is a sequence, with terms t1,
t2, t3…,tn,... then the expression of the form
t1 + t2 + t3 + …. + tn + … is
a series with terms t1, t2, t3……
Progression
It is not
necessary that the terms of a sequence always follow a certain pattern or they
are described by some explicit formula for the nth term. Those
sequences whose terms follow certain rule or follow certain pattern are called
progression. The special types of progression are
i) Arithmetic
progression
ii) Geometric
progression
iii) Harmonic
Progression.
Arithmetic Progression (A.P.)
A
sequence of numbers is said to be in arithmetic progression if the algebraic
differences between any term and preceding term is constant through the
sequence. Thus terms t1, t2, t3, ... are said
to be arithmetic progression if t2 - t1
= t3- t2 = t4 - t3 etc. The difference between any term
and its preceding term is called the common difference and it is denoted by d.
For example: 1, 3, 5, 7, … are in A.P.
Then the common difference = 3 - 1 = 5 - 3 = 2.
The nth term
Let a be the first term and d be the common difference of an A.P. Then
2nd term t2 = a + d = a + (2 - 1)d
3rd term t2 = a + 2d = a + (3 - 1) d
4th term t4 = a + 3d = a + (4 - 1) d
….. …. …..
nth tem tn = a + (n - 1) d
Arithmetic Mean
i) Single
arithmetic mean between a and b
Let
m be an arithmetic mean between a and b.
Therefore a, m, b are in A. P., then
m - a = b - m
or, 2m = a + b
Properties
of A.P.
i) If each of the
terms of an A.P. be increased or decreased by a constant quantity, the
resulting quantities are in A.P. with the same common difference as before.
ii) If each terms
of A.P. be multiplied by or divided by a constant quantity, the resulting
quantities are in A.P. with a common difference equal to that of given series
multiplied or divided by the corresponding constant quantity.
Geometric
Progression
A
sequence of numbers is said to be in geometric progression when the ratio of
each term to its preceding term is always a constant quantity. The constant
ratio is called the common ratio of G.P and it is denoted by r.
The nth
term
Let first
term of G.P. be a and r be the common ratio then,
Second term t2 = ar = ar2-1
Third term t3
= ar2 = ar3-1
Fourth term t4
= ar3 = ar4-1
… … …
nth
term tn = arn-1
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