What we are to prove is that for any positive integer n,

 is divisible by 3, that is, it is a multiple of 3.

For n=1

. 3 is a multiple of 3.

So it's true for n=1 

We must prove that if you know it's true for integer n=k, 
then it will be true for n=k+1.

Assume it's true for n=k.

 for some positive integer m

Multiply both sides by 

 for some positive integer m
 for some positive integer m



Write -4 as -1-3





 is a
multiple of 3.

Therefore since the theorem is true for n=k=1, it's true for n=k+1=2.

And since the theorem is true for n=k=2, it's true for n=k+1=3

Thus since the theorem is true for n=k=3, it's true for n=k+1=4

And this goes on forever, for we've proved it can't stop!

Edwin