Question 97923
We prove it through mathematical induction 

for n =1 
P(n)= 1^3 + 2^3 + 3^3
    = 36 which is divisble by 9

hence P(1) is true

Let it be true for n=m
P(m) is true 

We have to now prove that it is true for n=m+1



given P(m+1) = (m+1)^3 + (m+2)^3 + (m+3)^3
       =  (m+1)^3 + (m+2)^3 + m^3 + 9m^2 + 27m + 27 Expanding the last term
       = [ m^3 + (m+1)^3 + (m+2)^3 ] + 9m^2 + 27m + 27 Rearranging and grouping
       = P(m) + 9(m^2+ 3m +3)
the expression above is also divisible by 9 as P(m) is divisble by 9 and 
9(m^2+ 3m +3) is divisble by 9.

Hence it is true for all n . Proof by induction complete