Question

Prove by Induction
"The sum of the cube of any three consecutive integers is divisible by 9"

I confused with what is the basis step that I should take, since it says 'three consecutive integers'
I can prove it if it's non negative integers or positive integers, but it says 'integers' which means all integers.
or should I make it into cases
1. non negative integers
2. negative integers
3. the possibilities that the three consecutive integers are -2, -1,0 or -1,0,1

Answer 1

your case 1 and case 2 are identical. Since:\[(-(n+2))^3+(-(n+1))^3+(-n)^3=-(n^3+(n+1)^3+(n+2)^3)\] it follows that the sum of the cubes of three positive consecutive numbers (the right hand side) being divisible by 9 will imply that the sum of the cubes of three negative consecutive numbers (the left hand side) is also divisible by 9.

Answer 2

yes.., case 1 and case 2 are identical.. :)

can I prove it using this way?

Answer 3

yes that would be a valid proof.

Answer 4

ok.., thank you.. :)