Correct Answer - Option 4 : A rational number

**Concept:**

- A number r is called a rational number, if it can be written in the form
**p/q**, where** p and q are integers** and **q ≠ 0**.
- The
**decimal expansion** of a rational number is **either terminating** or **non-terminating recurring**.
- All the rational and irrational numbers make up the collection of
**real numbers**.

__Calculation:__

Let __\(x =n^7 + \dfrac{n^5}{5} + \dfrac{2n^3}{3} - \dfrac{n}{105}\)__

Where n = any positive integer

Let's take n = 1

⇒ \(x =1^7 + \dfrac{1^5}{5} + \dfrac{2\times 1^3}{3} - \dfrac{1}{105}\)

\(\Rightarrow x =1 + \dfrac{1}{5} + \dfrac{2}{3} - \dfrac{1}{105}\)

\(\Rightarrow x = \dfrac{13}{7}\)

We can see that **x is a rational number** & from the above options, it is clear that, only **option 4** belongs to the category of x.