Use mathematical induction to prove for all natural numbers [tex]n[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

P(x) = [tex]\frac{n^3}{3} + \frac{n^5}{5} + \frac{7n}{15} [/tex] is an integer.

Say [tex] S = \{ n\in N:P(x)\} [/tex]

Then [tex] 1\in S. [/tex] (1/3 + 1/5 + 7/15 = 15/15)

So assume for some [tex] k\in N, k\in S [/tex]

So, obviously I have to show [tex] (k+1)\in S [/tex].

I'm not sure how to start. In order for it to be an integer, then 15 must divide

[tex] 5n^3 + 3n^5 + 7n [/tex].

I've worked through, by substituting (k + 1), tried distributing throughout, but I can't seem to get anywhere to factor out a multiple of 15. Can anyone give a suggestion of how to manipulate this so I can prove it? Thanks

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# Induction problem

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