Question

For every positive integer $n,\frac{n^7}{7}+\frac{n^5}{5}+\frac{2n^3}{3}-\frac{n}{105}$ is

• A. an integer
• B. a rational number
• C. a negative real number
• D. an odd integer

Answer 1

The correct option is A an integer

$\frac{n^7}{7}+\frac{n^5}{5}+\frac{2n^3}{3}-\frac{n}{105}$

where $n$ is a positive integer.

When $n=1$, we get,

$\frac{n^7}{7}+\frac{n^5}{5}+\frac{2n^3}{3}-\frac{n}{105}$

$=\frac{15n^7+21n^5+70n^3-n}{105}$

$=\frac{15+21+70-1}{105}=\frac{105}{105}=1$ which is an integer.

When $n=2$,

we get;

$\frac{n^7}{7}+\frac{n^5}{5}+\frac{2n^3}{3}-\frac{n}{105}$

$=\frac{15n^7+21n^5+70n^3-n}{105}$

$=\frac{(15\times 128)+(21\times 32)+(70\times 28)-2}{105}$

$=\frac{1920+672+560-2}{105}$

$=\frac{3105}{105}=30$ which is an even positive integer.