Question

Prove that for all natural numbers n,1/5n^5+1/3n^3+7/15n is natural number

Answer 1

I have proved this result by
Mathematical induction in the
Attachment.
I hope this answer help you
.
.
Karups

Answer 2

Answer:

The given relation is proved below.

Step-by-step explanation:

Step 1:

Let p(n) denote the statement

$\frac{1}{5} n^{2}+\frac{1}{3} n^{3}+\frac{7}{15}$ n is a natural number.

Step 2:

$\begin{array}{l}{\mathrm{P}(1)=\frac{1}{5} \mathrm{x} 1^{2}+\frac{1}{3} \mathrm{x} 1^{3}+\frac{7}{15}} \\ {\mathrm{P}(1)=\frac{1}{5}+\frac{1}{3}+\frac{7}{15}} \\ {\mathrm{P}(1)=\frac{3+5+7}{15}} \\ {\mathrm{P}(1)=\frac{15}{15}}\end{array}$

Therefore 1 is a natural number.

So, p(1) is true.

Step 3:

So come to know that the result is true for n=k, that is p(k) is true.

Therefore we come to the conclusion.

$\frac{1}{5} \mathrm{k}^{2}+\frac{1}{3} \mathrm{k}^{3}+\frac{7}{15} \mathrm{k}$ is a natural number.