Suppose P0 P1 and P2 are all true And for all integers n gre

Suppose P(0). P(1) and P(2) are all true. And for all integers n greaterthanorequalto 0. if P(n), P(n + 1) and P(n + 2) are true then so is P(n + 3). For what n, can you be certain that P(n) is true?

Solution

   For all integers k>=0, if P(k) is true, then P(3k) is true
so only multiples of 0,1 and 2 can be proved true since they are generated from P(0), P(1) and P(2)
So P(0) generates only P(0), i.e. 3*0 is always 0.
P(1) generates P(3), P(9), P(27)...P(3*n) i.e. only integers that have a factor of 3.
P(2) generates P(6),P(18),..........P(6n) i.e. only integers that have a factor of 6.

Counter example is P(5) which cannot be generated by the given condition.