Question

Choose the following correct one which ${{\text{n}}^4}$ is less than for all ${\text{n}} \in {\text{N}}$.
a. ${10^{\text{n}}}$
b. ${4^{\text{n}}}$
c. ${10^{10}}$
d. None of the above

Answer 1

From the question, we have to choose the correct answer for which ${{\text{n}}^{\text{4}}}$ is less than for all ${\text{n}} \in {\text{N}}$. For the solution, we have to substitute the values in the given and compare them with the given options. Thus, we get the required answer.

Formula Used:
A natural number is an integer greater than $0$. The set of natural numbers is an infinite set containing the “counting numbers: $1,2,3,4,.........$” . The natural numbers start at $1$ and include all positive numbers without a fractional or decimal part. We use the symbol ${\text{N}}$ to refer to the natural number. Sometimes you will also see the natural numbers can be denoted as ${{\text{N}}^{\text{ + }}}$.

Complete step by step answer:
From the given, we have the mathematical expression ${{\text{n}}^{\text{4}}}$. Now, check whether the expression is less than the given options.
First, we have to choose ${\text{n}} = 1$,$1 \in {\text{N}}$. Then we get ${{\text{n}}^4} = {\left( 1 \right)^4} = 1$.

Option A: ${10^{\text{n}}}$ $\Rightarrow {10^1} = 10$ .
Option B: ${4^{\text{n}}}$ $\Rightarrow {4^1} = 4$ .
Thus, ${{\text{n}}^{\text{4}}}$ less than the other given three options.

Now, we have to choose${\text{n}} = 2$,$2 \in {\text{N}}$. Then we get ${{\text{n}}^4} = {\left( 2 \right)^4} = 16$.

Option A: ${10^{\text{n}}}$ $\Rightarrow {10^2} = 100$ .
Option B: ${4^{\text{n}}}$ $\Rightarrow {4^2} = 16$ .
Here ${{\text{n}}^{\text{4}}} = {4^{\text{n}}}$ but ${{\text{n}}^{\text{4}}} < {10^{\text{n}}}$.
There are so many differences for choosing the values of ${\text{n}}$.

Now, we are going to choose the values of ${\text{n}}$ to be large.
Let us take the value of $10 \in {\text{N}}$. Then we get ${{\text{n}}^4} = {\left( {10} \right)^4} = 10000$.

Option A : ${10^{\text{n}}}$ $\Rightarrow {10^{10}} = 10000000000$ .
Option B : ${4^{\text{n}}}$ $\Rightarrow {4^{10}} < {10^4}$ .
Here ${{\text{n}}^{\text{4}}} > {4^{\text{n}}}$ but ${{\text{n}}^{\text{4}}} < {10^{\text{n}}}$.
For the large values of ${\text{n}}$, ${10^{\text{n}}}$ is always less than ${{\text{n}}^{\text{4}}}$.

Hence, the correct answer is option (A).

Note: Trial and error is a method of reaching a correct solution or satisfactory result by trying out various means or theories until error is sufficiently reduced or eliminated. In other words, a way to solve things by making our best effort, seeing the result and how much it is in error, then making a better try until we get the desired result.