Question

For all $n\in N,thesumof\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$ is

• A. a negative integer
• B. a whole number
• C. a real number
• D. a natural number

Answer 1

Answer: (D)

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