Principle of Mathematical
Induction
Introduction:
Mathematical
induction is a mathematical proof technique. As a mathematical technique of
proving things, mathematical induction is essentially used to prove the
property of natural numbers.
The principle of mathematical
induction:
Statement:
Let
be a given statement where
is nature number such that
(i)
The
statement is true for
, i.e.,
is true.
(ii)
Let the
statement is true for then the statement is also true for
, where
is some positive integer.
Then is true for all natural numbers
.
Example: For all,
prove that
.
Proof:
Let be the given statement.
i.e.,.
For,
LHS
RHS
Hence
LHS = RHS.
So,
is true.
Let is true for some positive integer
, i.e.
Now
(Using (1))
Therefore
the statement is true for
, when it is for
.
Then by Principle of mathematical
induction
is true for all natural number
.
Hence.
(Proof)
Exercise 4.1 NCERT Book
Prove the following using the principle
of mathematical induction .
1.
Proof:
Let be the given statement.
i.e., .
For ,
LHS
RHS
Hence LHS = RHS.
So, is true.
Let is true for some positive integer
, i.e.
Now
(Using (1))
Therefore the statement is true for
, when it is true for
.
Then
by Principle of mathematical induction
is true for all natural number
.
Hence
(Proof)
2.
Proof:
Let
be the given statement.
i.e.,.
For ,
LHS
RHS
Hence LHS = RHS.
So, is true.
Let is true for some positive integer
, i.e.
Now
(Using (1))
Therefore the statement is true for
, when it is true for
.
Then by Principle of mathematical
induction
is true for all natural number
.
Hence (Proof)
3.
Proof:
Let be the given statement.
i.e.
For
,
LHS
RHS
Hence LHS = RHS.
So, is true.
Let is true for some positive integer
, i.e.
Now
(Using (1))
Therefore the statement is true for
, when it is true for
.
Then by Principle of mathematical
induction
is true for all natural number
.
Hence (Proof).
4.
Proof:
Let be the given statement.
i.e.
For ,
LHS
RHS
Hence LHS = RHS.
So, is true.
Let
is true for some positive integer
, i.e.
Now
(Using (1))
Therefore
the statement is true for
, when it is true for
.
Then by Principle of mathematical
induction
is true for all natural number
.
Hence (Proof).
5.
Proof:
Let
be the given statement.
i.e.
For ,
LHS
RHS
Hence LHS = RHS.
So, is true.
Let is true for some positive integer
, i.e.,
Now
(using (1))
Therefore
the statement is true for
, when it is true for
.
Then
by Principle of mathematical induction
is true for all natural number
.
Hence (Proof)
6.
Proof:
Let be the given statement.
i.e.
For
,
LHS
RHS
Hence
LHS = RHS.
So,
is true.
Let
is true for some positive integer
, i.e.
Now
(Using (1))
Therefore the statement is true for
, when it is true for
.
Then by Principle of mathematical induction
is true for all natural number
.
Hence ( Proof
7.
Proof:
Let
be the given statement.
i.e.
For
,
LHS
RHS
Hence
LHS = RHS.
So,
is true.
Let
is true for some positive integer
, i.e.
Now
(using (1))
Therefore the statement is true for
, when it is true for
.
Then by Principle of mathematical
induction is true for all natural number
.
Hence (Proof).
8.
Proof:
Let be the given statement.
i.e.
For,
LHS
RHS
Hence LHS = RHS.
So, is true.
Let is true for some positive integer
, i.e.
Now
(Using (1))
Therefore
the statement is true for
, when it is true for
.
Then by Principle of mathematical
induction
is true for all natural number
.
Hence (Proof).
9.
Proof:
Let be the given statement.
i.e.
For ,
LHS
RHS
Hence LHS = RHS.
So, is true.
Let is true for some positive integer
, i.e.
Now
(Using (1))
Therefore the statement is true for
, when it is true for
.
Then
by Principle of mathematical induction
is true for all natural number
.
Hence (Proof).
10.
Proof:
Let
be the given statement.
i.e.
For ,
LHS
RHS
Hence LHS = RHS.
So, is true.
Let is true for some positive integer
, i.e.
Now
(Using (1))
Therefore the statement is true for
, when it is true for
.
Then by Principle of mathematical
induction
is true for all natural number
.
Hence (Proof).
11.
Proof:
Let be the given statement.
i.e.
For
,
LHS
RHS
Hence
LHS = RHS.
So,
is true.
Let
is true for some positive integer
, i.e.
Now
(Using (1))
Therefore the statement is true for
, when it is true for
.
Then by Principle of mathematical
induction
is true for all natural number
.
Hence (Proof).
12.
is a multiple of
.
Proof:
Let be the given statement.
i.e., is a multiple of
.
For ,
, which is multiple of
So, is true.
Let is true for some positive integer
, i.e.
is a multiple of
Let
Now
(Using (1))
, which is multiple of
.
Therefore
the statement is true for
, when it is true for
.
Then
by Principle of mathematical induction
is true for all natural number
.
Hence is a multiple of (Proof).
13.
is multiple of
.
Proof:
Let be the given statement.
i.e., is multiple of
For ,
which is multiple of
So, is true.
Let is true for some positive integer
, i.e.,
is multiple of
, where
Now
(Using (1))
, which is divisible by
.
Therefore
the statement is true for
, when it is true for
.
Then
by Principle of mathematical induction
is true for all natural number
.
Hence is multiple of . (Proof).
14.
Proof:
Let be the given statement.
i.e.,
For ,
LHS
RHS
Since
So, is true.
Let is true for some positive integer
, i.e.,
Now
(Using (1))
Therefore
the statement is true for
, when it is true for
.
Then
by Principle of mathematical induction
is true for all natural number
.
Hence (Proof).