PMI- Principle of Mathematical Induction PMI - Proving by Mathematical Induction

1. MATHEMATICAL INDUCTION
PROVING BY MATHEMATICAL INDUCTION
PreCalculus

2. THE PRINCIPLE OF MATHEMATICAL
INDUCTION
•Let Sn be a statement for each positive integer
n. suppose that the following condition hold:
•1. S 𝟏 is true.
•2. If Sk is true, then Sk+1 should be true,
where k is any positive integer.
•Therefore, Sn is true for all positive integers n.

3. ILLUSTRATIVE EXAMPLES:
•Example 1: Using mathematical
induction, prove that
𝟏 + 𝟐 + 𝟑 + ⋯ + 𝒏 =
𝒏 𝒏+𝟏
𝟐
for all
positive integers n.

4. PMI
Step 1: n = 1
𝟏 =
𝟏 𝟏+𝟏
𝟐
=
𝟐
𝟐
= 1
• Example 1: Using mathematical induction,
prove that
𝟏 + 𝟐 + 𝟑 + ⋯ + 𝒏 =
𝒏 𝒏+𝟏
𝟐
for all positive integers
n.

5. PMI
Step 2: for n = k
𝟏 + 𝟐 + 𝟑 + ⋯ + 𝒌 =
𝒌 𝒌+𝟏
𝟐
and for n = k + 1
𝟏 + 𝟐 + 𝟑 + ⋯ + 𝒌 + (𝒌 + 𝟏) =
(𝒌 + 𝟏) 𝒌 + 𝟐
𝟐
• Example 1: Using mathematical induction,
prove that
𝟏 + 𝟐 + 𝟑 + ⋯ + 𝒏 =
𝒏 𝒏+𝟏
𝟐
for all positive integers
n.

6. PMI
and for n = k + 1
𝟏 + 𝟐 + 𝟑 + ⋯ + 𝒌 + (𝒌 + 𝟏) =
(𝒌 + 𝟏) 𝒌 + 𝟐
𝟐
𝒌 𝒌+𝟏
𝟐
+ (k + 1) =
(𝒌+𝟏) 𝒌+𝟐
𝟐
• Example 1: Using mathematical induction,
prove that
𝟏 + 𝟐 + 𝟑 + ⋯ + 𝒏 =
𝒏 𝒏+𝟏
𝟐
for all positive integers
n.

7. PMI
𝒌 𝒌+𝟏
𝟐
+ (k + 1) =
(𝒌+𝟏) 𝒌+𝟐
𝟐
𝒌 𝒌+𝟏 +𝟐(𝒌+𝟏)
𝟐
=
𝒌+𝟏 (𝒌+𝟐)
𝟐
=
(𝒌+𝟏) 𝒌+𝟐
𝟐
• Example 1: Using mathematical induction,
prove that
𝟏 + 𝟐 + 𝟑 + ⋯ + 𝒏 =
𝒏 𝒏+𝟏
𝟐
for all positive integers
n.

8. ILLUSTRATIVE EXAMPLES
•Example 2: Prove that
𝒊=𝟏
𝒏
𝟑𝒊 =
𝟑𝒏(𝒏 + 𝟏)
𝟐
.

9. EXAMPLE 3
Using mathematical induction, prove
that
𝟏 𝟐
+ 𝟐 𝟐
+ 𝟑 𝟐
+ ⋯ + 𝒏 𝟐
=
𝒏 𝒏+𝟏 (𝟐𝒏+𝟏)
𝟔
for
all positive integers n.

10. EXAMPLE 4
•Use mathematical induction to
prove that, for every positive
integer n, 7n – 1 is divisible by 6.

11. EXAMPLE 4
•Part 1
71 −1 = 6 = 6 · 1
71 − 1 is divisible by 6.

12. EXAMPLE 4
Part 2
Assume: 7k − 1 is divisible by 6.
To show: 7k+1 − 1 is divisible by 6.

13. Part 2
To show: 7k+1 − 1 is divisible by 6.
7k+1 − 1 = 7 · 7k − 1
= 6 · 7k + 7k −1
= 6 · 7k + (7k − 1)

14. To show: 7k+1 − 1 is divisible by 6.
By definition of divisibility, 6 · 7k is
divisible by 6. Also, by the hypothesis
(assumption), 7k − 1 is divisible by 6.
Hence, their sum (which is equal to
7k+1 − 1) is also divisible by 6.

15. EXAMPLE 5
•Use mathematical induction to
prove that, for every nonnegative
integer n, 3n-n+3 is divisible by
3.

16. EXAMPLE 6
•Use mathematical induction to
prove that 2n > 2n for every
integer n ≥ 3.

17. Reference:
• Aoanan, Grace O. et. al. (2018) General
Mathematics for
Senior HS, C&E Publishing, Inc.
• Garces, Ian June L. et. al. (2016) Precalculus
“Teaching
Guide for Senior High School, Commission on
Higher

18. THANK YOU
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