Use mathematical induction to prove that 5^(n) - 1 is divisible by four for all natural numbers n. Hint: if a number is divisible by 4, then it has a factor of 4. also, -1 = -5 +4
This is a take home test so I don't want the answer because I want to know how to do it.
this is what I have done so far.
1. prove P(1) is true:
5^(1) -1 = 4 which is divisible by four so this checks
2. prove P(k) is true:
5^(k) - 1 is divisible by 4
5^(k) - 1 = 4(m) where m is some number
3. P(k+1):
5^(k+1) - 1 =4( ____ )
from here I have no idea what to do..if someone could help me with a hint or something that would be great....
5^(k+1) - 1 =4(m+1)
To prove the statement using mathematical induction, here's the next step:
Step 3: Prove P(k + 1) is true.
Assuming that P(k) is true, that is:
5^k - 1 = 4m
Where m is some integer.
Now, we want to prove that P(k + 1) is true, which means we need to show that:
5^(k + 1) - 1 is divisible by 4.
To do that, let's manipulate the left side of the equation to see if we can express it as a multiple of 4:
5^(k + 1) - 1 = 5 * 5^k - 1
= 5 * (5^k - 1) + 4(5^k)
We can notice that 5 * (5^k - 1) is divisible by 4 since it has a factor of 4. Thus, we have:
5^(k + 1) - 1 = 5 * (5^k - 1) + 4(5^k)
= 4(5^k) + 4(5^k)
This equation shows that 5^(k + 1) - 1 is a multiple of 4 since it can be written as 4 multiplied by some integer.
Therefore, we have shown that if P(k) is true, then P(k + 1) is true, which completes the proof by mathematical induction.
Keep in mind the steps in mathematical induction: proving the base case (P(1)), assuming P(k) is true, and then proving P(k + 1) is true. This allows you to establish that the statement holds for all natural numbers.