I have to show that (n^5)/5+(n^3)/3+7n/15 is an integer for all n.
I have tried induction but after expanding the term I get something like 3(k^5)+15(k^4)+35(k^3)+45(k^2)+37k+15.
Any help?
What term did you expand?
Induction is a good idea: if you know that $\displaystyle f(n)$ is an integer (where $\displaystyle f(n)$ is your polynomial expression), then expand $\displaystyle f(n+1)-f(n)$. As you will see, what you get is a polynomial with integer coefficients, hence $\displaystyle f(n+1)-f(n)$ is an integer, and since $\displaystyle f(n+1)=f(n) + (f(n+1)-f(n))$ is the sum of two integers, it is an integer.