Question

If $P\left(n\right)=\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$, then statement "$P\left(n\right)$ is a natural number" is true,

• A. only for $n>1$.
• B. only for $n$ is an odd positive integer.
• C. only for $n$ is an even positive integer.
• D. $n\in N$

Answer 1

The correct option is D $n\in N$

Given $P\left(n\right)=\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$

for $n=1$ $P\left(1\right)=\frac{1}{5}+\frac{1}{3}+\frac{7}{15}=\frac{15}{15}=1$ (true)

for $n=2$ $P\left(2\right)=\frac{32}{5}+\frac{8}{3}+\frac{14}{15}=\frac{150}{15}=10$ (true )

$\therefore P\left(n\right)$ is true for $n=2$

Similarly by induction we say that $P\left(n\right)$ is a natural number $\forall nϵN$

Ans: D