Question

$\forall n\in N$, value of $\frac{n^4}{24}+\frac{n^3}{4}+\frac{11n^2}{24}+\frac{n}{4}$ is

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

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A a rational number
B an integer
C a natural number
D a real number

Given$\frac{[n^4+6n^3+11n^2+6n]}{24}$

$\frac{\left[n\right(n+1\left)\right(n+2\left)\right(n+3\left)\right]}{24}$

${=}^{(n+3)}{C}_4$

Thus the above number is always divisible by $24$ Thus all four options are correct.