6n −1 is divisibleby 5, for n >0 using mathematical induction
To prove that 6n − 1 is divisible by 5 for n > 0 using mathematical induction, we will follow these steps:
Step 1: Base case
For n = 1, we have 6n - 1 = 6(1) - 1 = 5, which is divisible by 5.
Step 2: Inductive hypothesis
Assume that 6k - 1 is divisible by 5 for some positive integer k > 0. This is our inductive hypothesis.
Step 3: Inductive step
We need to prove that 6(k + 1) - 1 is divisible by 5 based on our inductive hypothesis.
Simplifying 6(k + 1) - 1, we get:
6k + 6 - 1 = 6k + 5
Since we assumed that 6k - 1 is divisible by 5, we can write it as 5m for some positive integer m.
Therefore, 6k + 5 = 5m + 5 = 5(m + 1), which is divisible by 5.
Step 4: Conclusion
By mathematical induction, we have shown that if 6k - 1 is divisible by 5 for some positive integer k > 0, then 6(k + 1) - 1 is divisible by 5 as well.
Therefore, 6n - 1 is divisible by 5 for n > 0.
6n - 1 is divisible by 5 for n = 1
HOWEVER if n = 2
12 - 1 = 11
which is not divisible by 5 in my world.
Neither is 17