Mathematics - Mathematical Induction Question with Solution | TestHub

Let the statement $\mathrm{P}(\mathrm{n})$ be defined as $P(n):\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$ is a natural number for all $n\in N$ :
Step I : 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 $n=k$,
i.e. $\frac{k^5}{5}+\frac{k^3}{3}+\frac{7k}{15}=\lambda \in N$ $\dots$ (i)
Step III : For $\mathrm{n}=\mathrm{k}+1$,
$\frac{(\mathrm{k}+1{)}^5}{5}+\frac{(\mathrm{k}+1{)}^3}{3}+\frac{7(\mathrm{k}+1)}{15}$
$=\frac{1}{5}\left(k^5+5k^4+10k^3+10k^2+3k+1\right)$ $+\frac{1}{3}\left(k^3+3k^2+3k+1\right)+\frac{7}{15}k+\frac{7}{15}$
$=\left(\frac{k^5}{5}+\frac{k^3}{3}+\frac{7}{15}k\right)+\left(k^4+2k^3+3k^2+2k\right)$ $+\frac{1}{5}+\frac{1}{3}+\frac{7}{15}$
$=\lambda +{\mathrm{k}}^4+2{\mathrm{k}}^3+3{\mathrm{k}}^2+2\mathrm{k}+1$ [using equation (i)]
which is a natural number, since $\lambda \mathrm{k}\in \mathrm{N}$. 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 $\mathrm{n}$