I want to prove that
$\frac{n^3}{3}+\frac{n^5}{5}+\frac{7 n}{15}$
is an integer for every integer $n \geq 1$.
I define P(n) to be:
$\frac{n^3}{3}+\frac{n^5}{5}+\frac{7 n}{15}$ is an integer.
For my basis step, P(1) is true because
$\frac{1^3}{3}+\frac{1^5}{5}+\frac{7}{15}=1$ which is an integer.
The inductive step is what's tripping me up...
Let k be an arbitrary positive integer. Assume that P(k) is true, that is,
$\frac{k^3}{3}+\frac{k^5}{5}+\frac{7 k}{15}$ is an integer.
So based on that assumption, I need to now show that P(k+1) is true, i.e., that
$\frac{(k+1)^3}{3} +\frac{(k+1)^5}{5} +\frac{7 (k+1)}{15}$ is an integer.
At this point, I am stuck as to where to go next...
I have tried rewriting the assumption:
$\frac{k^3}{3}+\frac{k^5}{5}+\frac{7 k}{15}=15 m$ for some integer m. Then I solve for m:
$\frac{1}{15} \left(\frac{k^3}{3}+\frac{k^5}{5}+\frac{7 k}{15}\right)=m$
But this looks like a dead-end, seems there's nothing I can do with this to the "to prove" equation.
I have also tried re-writing the "to show" equation as this, but I get a dead end there and am not sure where to go next:
$\frac{1}{15} \left(5 (k+1)^3+3 (k+1)^5+7 (k+1)\right)$