1. Mathematical Induction
Chapter 3
Mathematical Induction

2. 1. The Principle of Mathematical Induction
Consider the following series
1 = 12
1 + 3 = 22
= 32
= 42
…. + (2n-1) = n2

3. Is it true when n = 100 ? 1. The Principle of Mathematical Induction
LHS = …. + (2(100)-1)
= …
= 10000
RHS = 1002
The proposition is true for n = 100.

4. Apply Mathematical Induction (M.I.) to prove the proposition
1. The Principle of Mathematical Induction
Is it true when n = ?
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 :
P(1) is true.
Assuming P(k) is true for any positive integer k, it can be proved that P(k + 1) is also true.

5. 1. The Principle of Mathematical Induction

6. For instance : it is a serious mistakes to prove the identity
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.

7. Prove by mathematical induction that
2. Some Simple Worked Examples
Prove by mathematical induction that
…. + (2n –1) = n2 for all positive integers.
Let P(n) be the proposition …. + (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 …. + (2k –1) = k2
When n = k + 1, RHS = (k + 1)2

8. LHS = 1+3+5+7+ …. +(2k – 1) + [2(k+1) －1]
2. Some Simple Worked Examples
LHS = …. +(2k – 1) + [2(k+1) －1] k2
= k k + 2 － = k2 + 2k + 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.

9. 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.

10. 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.

11. 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.

12. 3. Variations of the Method of Induction
(B) 2nd type of variation :

13. 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.