Principle of Mathematical Induction

Let the statement \(\mathrm{P}(\mathrm{n})\) be defined as
\(\mathrm{P}(\mathrm{n}): \frac{\mathrm{n}^5}{5}+\frac{\mathrm{n}^3}{3}+\frac{7 \mathrm{n}}{15}\) is a natural number for all \(\mathrm{n} \in \mathrm{N}\).
Step 1: For \(\mathrm{n}=1, \mathrm{P}(1): \frac{1}{5}+\frac{1}{3}+\frac{7}{15}=1 \in \mathrm{~N}\)
Hence, it is true for \(\mathrm{n}=1\).
Step II: Let it is true for \(\mathrm{n}=\mathrm{k}\),
i.e. \(\frac{\mathbf{k}^3}{5}+\frac{\mathbf{k}^3}{3}+\frac{7 \mathbf{k}}{15}=\lambda \in N\) ....(i)

Step III: For \(n=k+1\)

\(
\begin{aligned}
& \frac{(k+1)^5}{5}+\frac{(k+1)^3}{3}+\frac{7(k+1)}{15} \\
& =\frac{1}{5}\left(k^5+5 k^4+10 k^3+10 k^2+5 k+1\right) \\
& +\frac{1}{3}\left(k^3+3 k^2+3 k+1\right)+\frac{7}{15} k+\frac{7}{15} \\
& =\left(\frac{k^5}{5}+\frac{k^3}{3}+\frac{7}{15} k\right)+\left(k^4+2 k^3+3 k^2+2 k\right) \\
& +\frac{1}{5}+\frac{1}{3}+\frac{7}{15} \\
& =\lambda+k^4+2 k^3+3 k^2+2 k+1
\end{aligned}
\)

[using equation (i)]
which is a natural number, since \(\lambda k \in N\). Therefore, \(P(k+1)\) is true, when \(P(k)\) is true. Hence, from the principle of mathematical induction, the statement is true for all natural numbers n .