#1**+2 **

Show that it is true for n = 1

(1)^3 - 1 = 0 and this is divisible by 3

Assume that it is true for n = k

That is k^3 - k is divisible by 3

Note that we can write this as k (k^2 - 1) = k (k + 1) (k - 1) = (k - 1)(k)(k + 1)

Prove that it is true for k + 1

(k + 1)^3 - (k + 1) factor

(k + 1) [ (k + 1)^2 - 1]

( k + 1) ( k^2 + 2k + 1 - 1 ]

(k + 1) [ k^2 + 2k ]

(k + 1) [ k ( k + 2) ] =

k (k + 1) (k + 2) =

k(k + 1) ( k + 3 - 1) =

k(k + 1) ( k - 1 + 3) =

k(k + 1) (k - 1) + 3k(k+1)

(k-1) (k) (k + 1) + 3k(k + 1)

And we assumed that the first term was divisible by 3

And the second term is a multiple of 3, so it is also divisible by 3

So.... (k + 1)^3 - (k + 1) is divisible by 3

CPhill
Nov 26, 2018