To Prove
3n – 1 is a multiple of 2 ∀ n ∈ N
Proof
We can prove the given statement using many ways. We will discuss 2 of them.
Direct Proof
We know the product of two odd numbers is always an odd number. This implies that the term 3n is always odd ∀ n ∈ N. On subtracting an odd number from another odd number, we get an even number.
This implies that the term 3n – 1 is even since both 3n and 1 are odd numbers. Therefore, we can say that 3n – 1 is a multiple of 2 since all even numbers are multiples of 2. Hence Proved.
Mathematical Induction
Given
3n – 1 is a multiple of 2 ∀ n ∈ N
STEP 1
First, we will check if the statement is true for n = 1.
3n – 1 = 31 – 1 = 3 – 1 = 2
The statement is true for n = 1.
STEP 2
Let the statement be true for n = k. 3k – 1 is a multiple of 2. We can write it as
3k – 1 = 2x, ∀ x ∈ N ……..(1)
STEP 3
We need to check if the statement is true for n = k + 1 given that the statement is true for n = k.
………(2)
Substituting equation (1) in (2)
which is a multiple of 2.
Thus, the statement is true for n = k + 1 given that it is true for n = k. Therefore, by the principle of mathematical induction, the given statement “3n – 1 is a multiple of 2” is true for all n ∈ N.