\(\frac{k^5}{5}+\frac{k^3}{3}+\frac{7}{15 k}\) is if \(k \in \mathbf{N}\)
\(\frac{k^5}{5}+\frac{k^3}{3}+\frac{7}{15 k}\) is if \(k \in \mathbf{N}\)
A natural number
An integer
A positive fraction
Equal to zero
Solution
Let \(P(K)=\frac{k^5}{5}+\frac{k^3}{3}+\frac{7 k}{15}, k \in \mathbf{N}\)
\(\begin{gathered}
=\frac{3 k^5}{15}+\frac{5 k^3}{15}+\frac{(15-5-3) k}{15} \\
=\frac{3}{15}\left(k^5-k\right)+\frac{5}{15}\left(k^3-k\right)+k \\
=\frac{1}{5} \underbrace{k\left(k^2+1\right)(k+1)(k-1)}_{\text {divisible by } 5}+\frac{1}{3} \underbrace{k(k+1)(k-1)}_{\text {divisible by } 3}+k
\end{gathered}\)
So, overall it is a natural number.