Examples – Proof by Mathematical Induction for Divisibility Statements

Prove Divisibility Statements using Mathematical Induction examples with solutions

Question 1:

Prove the following statement by the principle of mathematical induction:

is divisible by 11.

Solution:

We can write,

P(n):  is divisible by 11.

Step 1:

We first show that the basis for induction P(1) is true, that is, P(n) is true for .

If , then,

,

which is divisible by 11.

Hence, P(1) is true, that is, P(n) is true for .

Step 2:

Assume that P(k) is true for some natural number k, that is,

is divisible by 11, or

, or where q is a natural number.

We need to prove that P(k +1) is also true. That is, we need to prove that is also divisible by 11 if is divisible by 11.

Now we have,

           (Because P(k) is true, and so )

,         (where 100q−9∈N, because q∈N)

which is divisible by 11.

Thus, P(k +1) is true whenever P(k) is true.

Therefore, from the principle of mathematical induction, the statement:

is divisible by 11,

is true for all natural numbers n.

Question 2:

Prove the following statement by the principle of mathematical induction:

is divisible by 36.

Solution:

We can write,

P(n):  is divisible by 36.

Step 1:

We first show that the basis for induction P(1) is true, that is, P(n) is true for .

If , then,

,

which is divisible by 36.

Hence, P(1) is true, that is, P(n) is true for .

Step 2:

Assume that P(k) is true for some natural number k, that is,

is divisible by 36, or

, or , where q is a natural number.

We need to prove that P(k +1) is also true. That is, we need to prove that is also divisible by 36 if is divisible by 36.

Now we have,

     (Because P(k) is true, and so )

,      (where 7q+k ∈ N, because q, k ∈ N)

which is divisible by 36.

Thus, P(k +1) is true whenever P(k) is true.

Therefore, from the principle of mathematical induction, the statement:

is divisible by 36,

is true for all natural numbers n.

Tags: mathematical induction examples with solutions, prove divisibility statements by induction

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