
Arithmetic Series
The series associated with an arithmetic sequence is known as an
arithmetic series. For example: 3 +
7 + 11 + 15 + 19 + 23 + 27 + 31 is the arithmetic series associated with the
arithmetic sequence: 3, 7, 11, 15, 19, 23, 27, 31.
The series associated with an arithmetic sequence is known as an
arithmetic series. For example: 3 +
7 + 11 + 15 + 19 + 23 + 27 + 31 is the arithmetic series associated with the
arithmetic sequence: 3, 7, 11, 15, 19, 23, 27, 31.
If three numbers a, b and c are in Arithmetic Sequence (AS) then
the middle term b is said to be the arithmetic
mean (AM) between the other two. For example, 10, 14 and 18 are in AS, so
14 is the arithmetic mean between 10 and 18. If any number of terms are in AS
then the terms between the first and the last terms are known as the arithmetic
means between the first and the last. For example, 11, 14, 17, 20, 23, 26, 29,
32 are in AS, so 14, 17, 20, 23, 26, 29 are the arithmetic means between 11 and
32.
Consider the following sequences.
a. 1, 6, 11, 16, ... … …
b. 6, 2, -2, -6, … … … …
c. 3, 4.5, 6, 7.5, … … …
In the first sequence, each term is increased by 5 than the preceding term. In the second sequence, each term is decreased by 4 than the
preceding term. Each term in the third sequence is increased by 1.5 than the
preceding term. In each of the above sequences, the difference between a term
and its preceding term is equal or constant. Such a sequence is said to be an
Arithmetic Sequence or Arithmetic Progression.
Let A = {2, 3, 4}, B = {6, 9, 12} and C = {a, b, c}. If f : A →
B and g : B → C are defined by f = {(2, 6), (3, 9), (4, 12)} and g = {(6, a),
(9, b), (12, c)},
f is the function from A to B such that
2 ∈ A, f(2) = 6 ∈ B
3 ∈ A, f(3) = 9 ∈ B
4 ∈ A, f(4) = 12 ∈ B
Range of f = {6, 9, 12} = domain of g.
Let A = {2, 3, 4}, B = {a, b, c} and the function f : A → B is
defined by f = {(2, a), (3, b), (4, c)}.
Here,
2 ∈ A, f(2) = a ∈ B
3 ∈ A, f(3) = b ∈ B
4 ∈ A, f(4) = c ∈ B
Also domain of f = {2, 3, 4} and range of f = {a, b, c}.
∴ f is one-to-one onto function.
Algebraic Functions are those functions which involve the sum or difference of algebraic
terms. The following are some different types of algebraic functions.
a. Constant funciton
b. Identity function
c. Linear function
d. Quadratic function
e. Cubic function
Consider two sets A and B. Any non-empty subset R of the
cartesian product A × B is called a relation from A to B. A special type of
relation associates each element of set A with the one and only element of B.
This is indeed a refinement of the concept of relation. Such a refinement is
known as a function. We may define a function in the following way:
A function from a set A to a set B is a relation (or rule) which
associates each element of A with a unique element of B.
Symbolically, we write
f : A → B
to mean “f is a function from A to B”. Further, an element y of
B associated with an element x of A is denoted by f(x). Equivalently, we write
y = f(x) which reads ‘y equals f of x’. Here f(x) is known as the image of f at
x or the value of f at x.
The letters f, g, h, F, G, H, φ etc. are reserved for denoting
functions.
If A and B are two non-empty sets then any subset of ordered pairs of a cartesian product A × B is called a relation from set A to set B. A relation is denoted by xRy or
simply R, if (x, y) ∈ R. A relation from set A
to A is called a relation on A.
If A and B are two non-empty sets then the set of all possible
ordered pairs (x, y) such that the first component x is an element of A and the
second component y is an element of B is called the Cartesian Product of set A
and B. It is denoted by A × B which is read as “A cross B”.
In the set-builder form, we can write
A × B = {(a, b): a ∈ A, b ∈ B}
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