<s>[INST] proof using the mathematical induction Prove $$\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}$$ is an integer for every positive integer k.
In proving for (n+1) integer,the expression is integer,I found $(n+1)^7$ term.I use binomial theorem to expand but finally it won't work(some term become integer While some other remaining as rational)
Any hint to solve the problem is appreciated.Thanks.
 [/INST] We have, for instance, $(k+1)^7/7 = (\sum_{i=0}^7 \binom{7}{i}k^i)/7$. Note that the only times the coefficient of $k^i$ in the numerator is not divisible by $7$ is when $i=0,7$. Therefore, $(k+1)^7/7 = k^7/7 + 1/7 + \text{integer}$. Applying the same logic to the other terms, we find
$$\begin{align}\frac{(k+1)^7}{7}+\frac{(k+1)^5}{5}+\frac{2(k+1)^3}{3}-\frac{k+1}{105}&=\left(\frac{k^7}{7}+\frac{k^5}{5}+\frac{2k^3}{3}-\frac{k}{105}\right) \\&+\left(\frac{1}{7}+\frac{1}{5}+\frac{2}{3}-\frac{1}{105}\right) + \text{integer}\\ &= \text{integer} + 1 +\text{integer} \\&= \text{integer}.\end{align}$$
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