Section 8.4
Mathematical
Induction
Mathematical Induction
In this section we are going to perform a type of
mathematical proof called mathematical induction.
This is used to show that a formula used to find the
sum of a series is true.
This type of proof requires a 3-step approach.
Step 1: Begins with the word SHOW
Step 2: Begins with the word ASSUME
Step 3: Begins with the work PROVE
Mathematical Induction
Before we begin, one needs to be able to find Sk+1
based on Sk.
Example:
Solution:
Sk = 3k + 2, find Sk+1
Substitute k + 1 in for k found in Sk.
Sk 1  3  k  1  2
 3k  3  2
Sk 1  3k  5
Mathematical Induction
Now you try:
Given Sk, find Sk+1 for each of the following:
1. Sk  k  k  1
S k 1   k  1 k  1  1
  k  1 k  2 
3. Sk  k  k  1 k  2
S k 1   k  1 k  1  1 k  1  2 
  k  1 k  2  k  3
2. S k 
S k 1 
2k
3k  1
2  k  1
3  k  1  1
2k  2
3k  3  1
2k  2

3k  4

Mathematical Induction
Now, let’s begin proving by mathematical induction.
Prove: Sn  1  2  3  4  ...  n 
Step 1: Show S1 = 1 is true.
11  1
S1 
2
1 2 

2
2
 1
2
TRUE
n  n  1
2
This means to substitute 1 into
the formula part and show that
you get the first term of the
series which would be the sum
of the first term.
Mathematical Induction
Prove: Sn  1  2  3  4  ...  n 
Step 2: Assume Sk  1  2  3  4  ...  k 
Simply replace n in the
Prove statement with k.
n  n  1
2
k  k  1
2
Mathematical Induction
Prove: Sn  1  2  3  4  ...  n 
n  n  1
2
Step 3: Prove S k 1  1  2  3  4  ...  k   k  1
k  1 k  1  1


S k 1  1  2  3  4  ...  k   k  1 
Go back to the Assume
Statement add k + 1 as
the next term of the
sequence and replace
all k’s in the sum formula
with k + 1. Then simplify
the formula.
2
 k  1 k  2 
2
Mathematical Induction
Prove: Sn  1  2  3  4  ...  n 
n  n  1
2
Do the Proof:
Write the first part of the
Sk 1  1  2  3  4  ...  k   k  1
prove statement
k  k  1
Remove 1 + 2 + 3 + 4 + … + k and replace it

  k  1
2
with the “formula piece” from the Assume
Statement and add on k +1.
k  k  1  2  k  1

Use your algebra skills to simplify. This should
2
agree with the first part of the prove statement.
k  1 k  2 

 S k 1 
Q.E.D by mathematical induction
2
Mathematical Induction
Let’s work this one together:
Prove: Sn  1  3  5  7  ...  (2n  1)  n
Make sure you get it in your notes to assist with the homework!
2
Mathematical Induction
Now you try this one:
Prove: Sn  1  2  3  4  ...  n 
2
2
2
2
2
n  n  1 2n  1
6
Mathematical Induction
What you should know:
How to do a mathematical prove
using mathematical induction!