Question

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

• A. an integer
• B. a natural number
• C. a positive fraction
• D. none of these

Answer 1

The correct option is B a natural number

Given expression
$\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$

Put $n=1$
$\frac{1}{5}+\frac{1}{3}+\frac{7}{15}$ $=1$
which is a natural number, integer, a positive fraction.
Put $n=2$
$\frac{2^5}{5}+\frac{2^3}{3}+\frac{14}{15}$ $=10$
which is a natural number , integer , a positive fraction.
Also, $P\left(3\right)=50$ and $P\left(4\right)=212$
So, P(3) and P(4) are also natural number , integer , positive fraction
Now, let $P\left(n\right)$ is true
$\frac{n^5}{5}+\frac{n^3}{3}+\frac{7}{15n}$ is a natural number (As a natural number is integer as well as a positive fraction)

We will check for $P(n+1)$
Consider, $\frac{(n+1{)}^5}{5}+\frac{(n+1{)}^3}{3}+\frac{7(n+1)}{15}$
$=\frac{n^5+5n^4+10n^3+10n^2+5n+1}{5}+\frac{n^3+3n^2+3n+1}{3}+\frac{7n+7}{15}$
$=(\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15})+(n^4+2n^3+3n^2+2n)+(\frac{1}{5}+\frac{1}{3}+\frac{7}{15})$
$=\text{a natural number}$
Hence, $P(n+1)$ is true.
So, by mathematical induction method, we can say that given expression is a natural number for all $n\in N$