Question

Which one of the following is true?
A. $\left( {{\text{1 + }}\dfrac{{\text{1}}}{{\text{n}}}} \right){\text{ < }}{{\text{n}}^{\text{2}}}$, n is a positive integer.
B. $\left( {{\text{1 + }}\dfrac{{\text{1}}}{{\text{n}}}} \right){\text{ < 2}}$, n is a positive integer.
C. $\left( {{\text{1 + }}\dfrac{{\text{1}}}{{\text{n}}}} \right){\text{ < }}{{\text{n}}^3}$, n is a positive integer.
D. $\left( {{\text{1 + }}\dfrac{{\text{1}}}{{\text{n}}}} \right){\text{ }} \geqslant {\text{2}}$, n is a positive integer.
E. $\left( {{\text{1 + }}\dfrac{{\text{1}}}{{\text{n}}}} \right){\text{ = }}{{\text{n}}^{\text{2}}} + n + 3$, n is a positive integer.

Answer 1

In the question, to get the correct answer, we have to proceed by going through the options. Using the relation given, we will solve the particular option and then after take the next option and check it into the corresponding mathematical expression and then check whether it is correct or not.

Complete step-by-step answer:
First of all, we will write the given relation:
$\left( {{\text{1 + }}\dfrac{{\text{1}}}{{\text{n}}}} \right){\text{ < }}{{\text{n}}^{\text{2}}}$, n is a positive integer.
Take any value of integer for given n.
Let, n=1
Put, this value and check the given relation:
$\Rightarrow$ $\left( {{\text{1 + }}\dfrac{{\text{1}}}{{\text{n}}}} \right){\text{ < }}{{\text{n}}^{\text{2}}}$
$\Rightarrow$2 < 1
So, this is not a correct relation.
Let us take Second option:
$\Rightarrow$ $\left( {{\text{1 + }}\dfrac{{\text{1}}}{{\text{n}}}} \right){\text{ < 2}}$
Take any value of integer for given n.
Let, n=1
Put, this value and check the given relation:
$\Rightarrow$2 < 2
So, this is not a correct relation.
Let us take third option:
$\Rightarrow$ $\left( {{\text{1 + }}\dfrac{{\text{1}}}{{\text{n}}}} \right){\text{ < }}{{\text{n}}^3}$
Take any value of integer for given n.
Let, n=1
Put, this value and check the given relation:
$\Rightarrow$2 < 1
So, this is not a correct relation.
Let us take fourth option:
$\Rightarrow$ $\left( {{\text{1 + }}\dfrac{{\text{1}}}{{\text{n}}}} \right){\text{ }} \geqslant {\text{2}}$
Take any value of integer for given n.
Let, n=1
Put, this value and check the given relation:
$\Rightarrow$2 $\geqslant$2
So, this is a correct relation.
Let us take fifth option:
$\Rightarrow$ $\left( {{\text{1 + }}\dfrac{{\text{1}}}{{\text{n}}}} \right){\text{ = }}{{\text{n}}^{\text{2}}}{\text{ + n + 3}}$
Take any value of integer for given n.
Let, n=1
Put, this value and check the given relation:
$\Rightarrow$2 = 5
So, this is not a correct relation.
Thus, by analysis of all the given options, we concluded that the fourth option is a correct relation.
Therefore, option (D) is the correct answer.

Note: In this question, we will need to know the basics of integers. As we know Integers are a bigger collection of numbers that include whole numbers, negative numbers, and zero. Or we say an integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples of integers are: -5, 1, 5, 8.