First of all we have that
Now
proof by induction
n=1
and 0 is divisable by 9
assume n=k
is divisable by 9
Show k+1
Now we want to use the induction hypothesis so we will fix up the equation so we can
so the first term is divisable by 9 by the induction hypothesis and 9k is obviously divisable by 9 so we are done.
I believe that this proof could be approached a couple of different ways. Below I use modular arithmatic and the idea of congruency for the proof.
PF: Then n must be congruent to one of the following
0 , 1, 2, 3, 4, 5, 6, 7, 8 (mod 9).
In all cases,
--------------------------------
Josh
--------------------------------