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 inductive 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.
5^n - 1 is divisible by4
Step 1
We begin by verifying that the statement is true for n= 1.
5^1 - 1 = 5-1 = 4
Since 4 is divisible by 4, 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
5^k - 1 is divisible by4.
Using the definition of divisibility, we have that the statement above implies that there is a natural number p such that 5^k - 1 = 4* p.
Step 3
Here, we have to show that the statement is true for n= k+1.
5^(k+1) - 1 is divisible by4
To prove this, we start with the inductive hypothesis and perform some operations to it.
Since p is a natural number, 5p + 1 is a natural number as well.
4(5p + 1) is divisible by 4.
In consequence, 5^(k+1)-1 is divisible by 4, which proves that 5^n-1 is divisible by 4 for n=k+1. As such, the given statement is true for all positive integers.
