Prove via Mathematical Induction that
(7*n) - 1 is divisible by 6.
I have that it is divisble when n=1, but not n=2 and so on. How should I write the inductive proofs?
It's obviously not true.
7*2-1 = 13 which is not divisible by 6
I think you mean 7^n-1 is divisible by 6
when n=1, 7^1-1 = 7-1 = 6 is divisible by 6
assume for n=k
when n=k+1,
7^(k+1)-1 = 7*7^k - 1
= 7*7^k -7 + 6
= 7(7^k-1) + 6
but, 7^k-1 is divisible by 6, and 6 is divisible by 6, so 7(7^k-1)+6 is divisible by 6
since 6a+6b = 6(a+b)
To prove that (7n) - 1 is divisible by 6 using mathematical induction, you need to follow these steps:
Step 1: Base case
Start by proving that the statement holds for the base case, which is typically n = 1. In this case, substitute n = 1 into the expression (7n) - 1:
(7 * 1) - 1 = 7 - 1 = 6, which is divisible by 6. Therefore, the statement holds true for the base case.
Step 2: Inductive hypothesis
Assume that the statement holds true for some positive integer k, which means that (7k) - 1 is divisible by 6.
Step 3: Inductive step
Now, you need to prove that the statement holds for the next positive integer, k + 1. Substitute n = k + 1 into the expression (7n) - 1 and simplify it:
(7 * (k + 1)) - 1 = 7k + 7 - 1 = (7k) + 6
Step 4: Applying the inductive hypothesis
Using the inductive hypothesis, we know that (7k) - 1 is divisible by 6. Therefore, we can rewrite (7k) + 6 as (6m) + 6, where m is some integer. Simplifying further, we get:
(6m) + 6 = 6(m + 1)
So, (7 * (k + 1)) - 1 can be expressed as 6(m + 1), which is divisible by 6.
Step 5: Conclusion
By proving that the statement holds for the base case (n = 1) and showing that if it holds for k, then it holds for k + 1, you have proven that (7n) - 1 is divisible by 6 for all positive integers n by mathematical induction.
To prove that (7n) - 1 is divisible by 6 for all positive integers n using mathematical induction, you need to follow three steps:
1. Base case: Start by proving the statement for the base case, which is usually n = 1. Show that when n = 1, the expression (7n) - 1 is divisible by 6.
Substituting n = 1 into the expression, we get:
(7 * 1) - 1 = 7 - 1 = 6
6 is divisible by 6, so the base case holds.
2. Inductive hypothesis: Assume that the statement holds true for some positive integer k. That is, assume that (7k) - 1 is divisible by 6.
3. Inductive step: Prove that the statement also holds true for the next positive integer, k + 1. Substitute n = k + 1 into the expression (7n) - 1 and try to prove that it is divisible by 6.
(7 * (k+1)) - 1 = (7k + 7) - 1 = 7k + 6
Now, using the inductive hypothesis, we know that (7k) - 1 is divisible by 6. So, we can write (7k) - 1 as 6m for some positive integer m.
(7k) - 1 = 6m
Substituting this into the expression obtained from the inductive step:
7k + 6 = 6m + 7
Rearranging the terms, we get:
7(k + 1) - 1 = 6m + 6 + 1 = 6(m + 1)
From this equation, we can see that (7(k + 1)) - 1 is also divisible by 6. So, the inductive step is verified.
Therefore, by mathematical induction, we have proven that (7n) - 1 is divisible by 6 for all positive integers n.