Question

Prove the following using mathematical induction
For all positive integers n, 3|(n^3 - n)

Answer 1

is that 3 times the absolute value of n^3-n ?

Answer 2

\[3|(n^3-n)=> 4 k=n^3-n \text{ for some integer k}\]
show it is true for n=1:
\[4k=1^3-1=0 \text{ which is true since 4k=0 for some integer k}\]
now assume it is true for n=k

Answer 3

now show it is true for n=k+1

Answer 4

and lets choose a different integer besides n=k
since we already use k once
how about for some integer n=c
and now show it is true for n=c+1

Answer 5

(n^3 - n) = n (n+1) (n-1)