Proof by Induction
STEP 1
For n = 1:
The statement is true for n = 1
STEP 2
Assume the statement is true for n = k:
So, for n = k + 1:
If the statement is true for n = k, then it is true for n = k + 1
STEP 3
Based on step 1 and 2, then:
The statement is true for n = 1, then the statement is true for n = 2
The statement is true for n = 2, then the statement is true for n = 3
and so on,
The statement is true for n ≥ 1
Exercise
Proof the statements below by mathematical induction
Solution:
STEP 1
For n = 1:
The statement is true for n = 1
STEP 2
Assume the statement is true for n = k:
So, for n = k + 1:
If the statement is true for n = k, then it is true for n = k + 1
STEP 3
Based on step 1 and 2, then:
The statement is true for n = 1, then the statement is true for n = 2
The statement is true for n = 2, then the statement is true for n = 3
and so on,
The statement is true for n ≥ 1
Solution:
STEP 1
For n = 1:
The statement is true for n = 1
STEP 2
Assume the statement is true for n = k:
So, for n = k + 1:
If the statement is true for n = k, then it is true for n = k + 1
STEP 3
Based on step 1 and 2, then:
The statement is true for n = 1, then the statement is true for n = 2
The statement is true for n = 2, then the statement is true for n = 3
and so on,
The statement is true for n ≥ 1
Solution:
STEP 1
For n = 1:
The statement is true for n = 1
STEP 2
Assume the statement is true for n = k:
So, for n = k + 1:
If the statement is true for n = k, then it is true for n = k + 1
STEP 3
Based on step 1 and 2, then:
The statement is true for n = 1, then the statement is true for n = 2
The statement is true for n = 2, then the statement is true for n = 3
and so on,
The statement is true for n ≥ 1
is divisible by 4, for
Solution:
STEP 1
For n = 1:
is divisible by 4
The statement is true for n = 1
STEP 2
Assume the statement is true for n = k:
So, for n = k + 1:
If the statement is true for n = k, then it is true for n = k + 1
STEP 3
Based on step 1 and 2, then:
The statement is true for n = 1, then the statement is true for n = 2
The statement is true for n = 2, then the statement is true for n = 3
and so on,
The statement is true for n ≥ 1
is divisible by 5, for
Solution:
STEP 1
For n = 1:
is divisible by 5
The statement is true for n = 1
STEP 2
Assume the statement is true for n = k:
So, for n = k + 1:
If the statement is true for n = k, then it is true for n = k + 1
STEP 3
Based on step 1 and 2, then:
The statement is true for n = 1, then the statement is true for n = 2
The statement is true for n = 2, then the statement is true for n = 3
and so on,
The statement is true for n ≥ 1
, for
Solution:
STEP 1
For n = 2:
The statement is true for n = 2
STEP 2
Assume the statement is true for n = k:
So, for n = k + 1:
Since and , then the statement is true for n = k + 1
STEP 3
Based on step 1 and 2, then:
The statement is true for n = 2, then the statement is true for n = 3
The statement is true for n = 4, then the statement is true for n =5
and so on,
The statement is true for n > 1
Solution:
STEP 1
For n = 1:
The statement is true for n = 1
STEP 2
Assume the statement is true for n = k:
So, for n = k + 1:
If the statement is true for n = k, then it is true for n = k + 1
STEP 3
Based on step 1 and 2, then:
The statement is true for n = 1, then the statement is true for n = 2
The statement is true for n = 2, then the statement is true for n = 3
and so on,
The statement is true for n ≥ 1