2011-04-19, 12:48 AM
Base case: Prove it's true for (-1, 0, 1)
Sum of their cubes is... hey presto: 0. Which is to some extent divisible by 9.
Induction: Prove if it's true for n, then it's true for n+1
n: [n^3 + (n+1)^3 + (n+2)^3] has a factor of 9
n+1: [(n+1)^3 + (n+2)^3 + (n+3)^3]
So you have to prove that if the first equation is divisible by 9, so is the second one - probably by demonstrating that when you subtract one from the other, the difference is divisible by 9.
Sum of their cubes is... hey presto: 0. Which is to some extent divisible by 9.
Induction: Prove if it's true for n, then it's true for n+1
n: [n^3 + (n+1)^3 + (n+2)^3] has a factor of 9
n+1: [(n+1)^3 + (n+2)^3 + (n+3)^3]
So you have to prove that if the first equation is divisible by 9, so is the second one - probably by demonstrating that when you subtract one from the other, the difference is divisible by 9.