## Solve Lesson 5 Page 29 Math Study Topic 10 – Kite>

Topic

Prove that for all $$n \in \mathbb{N}*$$, we have:

a) $${13^n} – 1$$ is divisible by 6.

b) $${4^n} + 15n – 1$$ is divisible by 9.

Solution method – See details

Prove that the statement is true for $$n \ge p$$ then:

Step 1: Check the statement is true for $$n = p$$

Step 2: Suppose the proposition is true for natural numbers $$n = k \ge p$$ and prove the statement true for $$n = k + 1.$$ Conclusion.

Detailed explanation

a)

Step 1: When $$n = 1$$ we have $${13^1} – 1 = 12$$ divisible by 6.

So the statement is true for $$n = 1$$

Step 2: With k being an arbitrary positive integer whose proposition is true, we have to prove the proposition to be true for k + 1, that is:

$${13^{k + 1}} – 1$$ is divisible by 6.

Indeed, by the assumption of induction we have:

$${13^k} – 1$$ is divisible by 6.

I guess

$${13^{k + 1}} – 1 = {13.13^k} – 1 = 13.\left( {{{13}^k} – 1} \right) + 12$$ divisible by 6

So the statement is true for k+1. Thus, according to the principle of mathematical induction, the statement is true for all $$n \in \mathbb{N}*$$.

b)

Step 1: When $$n = 1$$ we have $${4^1} + 15.1 – 1 = 18$$ divisible by 9.

So the statement is true for $$n = 1$$

Step 2: With k being an arbitrary positive integer whose proposition is true, we have to prove the proposition to be true for k + 1, that is:

$${4^{k + 1}} + 15.(k + 1) – 1$$ is divisible by 9.

Indeed, by the assumption of induction we have:

$${4^k} + 15k – 1$$ is divisible by 9.

I guess

$${4^{k + 1}} + 15.(k + 1) – 1 = {4.4^k} + 15k + 14 = 4\left( {{4^k} + 15k – 1} \right) – 45k + 18$$ divisible by 9

So the statement is true for k+1. Thus, according to the principle of mathematical induction, the statement is true for all $$n \in \mathbb{N}*$$.