Can anyone help me get started with this proof using induction? Thank a lot everyone.
Prove that the product of any three consecutive positive integers is divisible by 6.
Base case so the proposition is true for the consecutive numbers .
Suppose the proposition is true for three consecutive numbers starting from .
Then divides , now consider the product of three consecutive integers starting for :
but divides and as one of and must be even divides , and so we conclude that divides .
Now we have proven the base case and that if the proposition is true for three consecutive integers starting from it is true for three consecutive integers starting from , and so by induction it is true for every set of
three consective positive integers.
A slight variation of CaptainBlack's proof.
We want to proveProve that the product of any three consecutive positive integers is divisible by 6.
Verify . . . True!
Assume for some integer
Add to both sides:
The left side is: .
. . Factor: .
The right side is: .
. . is the product of two consecutive integers.
. . Hence, it is divisible by 2:.
The right side is:.
From [L] and [R],we have: .
We have established ..The inductive proof is complete.