Use the fact that when a is divisible by b, it means that there is a natural number r such that a = b* r. Use this to write an equation from the nductive hypothesis.
See solution.
Practice makes perfect
Let's begin by recalling that when a is divisible by b, it means that there is a natural number r such that a = b* r. We will use this and mathematical induction to prove that the statement below is true for all positive integers.
7^n - 1 is divisible by6.
Step 1
We begin by verifying that the statement is true for n= 1.
7^1 - 1 = 7-1 = 6
Since 6 is divisible by 6, we have that the statement is true for n=1.
Step 2
Next, we assume that the given statement is true for a natural number k.
Inductive Hypothesis
7^k - 1 is divisible by6
Using the definition of divisibility, we have that the statement above implies that there is a natural number p such that 7^k - 1 = 6* p.
Step 3
Here, we have to show that the statement is true for n= k+1.
7^(k+1) - 1 is divisible by6
To prove this, we start with the inductive hypothesis and perform some operations to it.
Since p is a natural number, 7p+1 is a natural number as well.
6(7p+1) is divisible by 6
In consequence, 7^(k+1)-1 is divisible by 6, which proves that 7^n-1 is divisible by 6 for n=k+1. As such, the given statement is true for all positive integers.
