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

Answer: (A)

an integer

Let P(n) =$\frac{n^7}{7}+\frac{n^5}{5}+2n^{3}3-\frac{n}{105}$
P(1) = $\frac{1}{7}+\frac{1}{5}+\frac{2}{3}-\frac{1}{105}$ = 1 (integer)
P(2) = $2^4(\frac{8}{7}+\frac{2}{5}+\frac{1}{3})-\frac{2}{105}$ = 15 (integer) etc.
Hence P(n) is an integer.