# How to prove that the fraction is integer by mathematical induction?

How can I prove that the fraction

$$P(n) = \frac{1}{60} (7 * 4^n - 42 * (-1)^n - 10)$$

is integer for all natural numbers by mathematical induction method?

I tried to find the difference between $P(k + 1)$ and $P(k)$, but could not transform it to the original fraction.

• Show that $60$ divides the expression by induction. Did you try already $n=1$? The induction has been done several times here, in several variations, see for example here. Mar 17, 2018 at 19:09
• Write $P(n)=Q(n)/60.$ Show that $Q(n+1)-Q(n)$ is divisible by $60$. Mar 17, 2018 at 19:14
• Hint. it may be easier to use the difference $P(k+2) - P(k)$. Mar 17, 2018 at 19:20

$$P(1)=\frac{1}{60}\left(28+42-10\right)=1$$ $$P(2)=\frac{1}{60}\left(112-42-10\right)=1$$ $$P(k+2)-P(k)=\frac{4^k\times 105}{60}=7\times4^{k-1}$$
• how is $4^k * 112$ divisible by $60$?
• Make it $7 * 4^{k-2}$