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

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For every positive integer n, $n^7 + \dfrac{n^5}{5} + \dfrac{2n^3}{3} - \dfrac{n}{105}$ is:
1. An odd integer
2. An integer
3. A negative real number
4. A rational number

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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.