Chapter 3
Mathematical Induction
1. The Principle of Mathematical Induction
Consider the following series
1 = 12
1 + 3 = 22
1 + 3 + 5 = 32
1 + 3 + 5 + 7 = 42
1 + 3 + 5 + 7 + …. + (2n-1) =
2
n
1. The Principle of Mathematical Induction
Is it true when n = 100 ?
When n = 100
LHS = 1 + 3 + 5 + 7 +…. + (2(100)-1)
= 1 + 3 + 5 + 7 + …. + 199
= 10000
RHS = 1002
= 10000
The proposition is true for n = 100.
1. The Principle of Mathematical Induction
Is it true when n = 100000 ?
Apply Mathematical Induction
(M.I.) to prove the proposition
A proposition P(n) is true for all positive integers
n if both of the following conditions are satisfied :
1. P(1) is true.
2. Assuming P(k) is true for any positive
integer k, it can be proved that P(k + 1) is
also true.
1. The Principle of Mathematical Induction
1. The Principle of Mathematical Induction
Note :
Mathematical induction cannot be used to
prove whose variables are not positive
integers.
For instance : it is a serious mistakes to
prove the identity
x3 – 1 = (x - 1)(x2 + x + 1), for all xR.
2. Some Simple Worked Examples
Prove by mathematical induction that
1 + 3 + 5 + …. + (2n –1) = n2 for all positive integers.
Let P(n) be the proposition 1 + 3 + 5 + 7 + …. + (2n –1) = n2
When n = 1, RHS = 12 = 1
LHS = 1
P(1) is true.
Assume P(k) is true for any positive (+ve) integer k.
i.e. 1 + 3 + 5 + 7 + …. + (2k –1) = k2
When n = k + 1, RHS = (k + 1)2
2. Some Simple Worked Examples
LHS = 1+3+5+7+ …. +(2k – 1) + [2(k+1) －1]
k2
= k2
= k2 + 2k + 1
+ 2k + 2 －1
= (k + 1)2
∴ P(n) is true for n = k + 1 if n = k is true .
By M.I., P(n) is true for all +ve integers n.
3. Variations of the Method of Induction
(A) 1st type of variation :
Let P(n) be a proposition involving
positive integer n.
If (i) P(n) is true for n = 1 and n = 2
and (ii) if P(n) is true for some positive
integers k and k + 1,then
P(n) is also true for n = k + 2,
then P(n) is true for all positive integers n.
3. Variations of the Method of Induction
(A) 1st type of variation :
Note :
The principle may be applied
to the proposition of the form
an - bn or an + bn.
3. Variations of the Method of Induction
(B) 2nd type of variation :
Let P(n) be a proposition involving integer n.
If (i) P(n) is true n = ko,where ko is an integer
not necessarily equals 1, and
(ii) if P(n) is true for n = k (k  k0) then P(n)is
also true for n = k + 1.
then P(n) is true for all integers n  ko.
3. Variations of the Method of Induction
(B) 2nd type of variation :
e.g. Prove that for every positive integer
n  5, 2  n .
n
2
(i ) When n  5, LHS  2  32,
5
RHS  5  25
2
i.e. 2  n is true for n  5.
n
2
3. Variations of the Method of Induction
(C) 3rd type of variation :
Let P(n) be a proposition involving integer n.
If (i) P(n) is true for n = 1 and n = 2,
and (ii) if P(n) is true for some positive
integer k, then P(n) is also true for n = k + 2,
then P(n) is true for all positive integers n.