Question

For positive integer n, ${10^{n - 2}} > 81n$ when
A. $n < 5$
B. $n > 5$
C. $n \geqslant 5$
D. $n > 6$

Answer 1

Hint: In a question like this where comparing or solving the inequalities is not possible we follow the hit and trial method for this question, in this method the equation or the inequality is checked of the initial values of the variable, and after we’ll be able to conclude the result on the base of those initial values.

Complete step-by-step answer:
Given data: ${10^{n - 2}} > 81n$
In the given inequality we cannot simply compare the expression as one is a multiple of n and the other is exponential, so we will follow the hit and trial method for this question,
As n is given a positive integer
For $n = 1$ i.e. substituting the value of n as 1
 $ \Rightarrow {10^{ - 1}} > 81$ , which is not true as $\left( {0.1 < 81} \right)$
For $n = 2$ i.e. substituting the value of n as 2
 $ \Rightarrow {10^0} > 162$ , which is not true as $\left( {1 < 162} \right)$
For $n = 3$ i.e. substituting the value of n as 3
 $ \Rightarrow {10^1} > 243$, which is not true as $\left( {10 < 243} \right)$
For $n = 4$ i.e. substituting the value of n as 4
 $ \Rightarrow {10^2} > 324$ , which is not true as $\left( {100 < 324} \right)$
For $n = 5$ i.e. substituting the value of n as 5
 $ \Rightarrow {10^3} > 405$, which is true as $\left( {1000 > 405} \right)$
For $n = 6$ i.e. substituting the value of n as 6
 $ \Rightarrow {10^4} > 486$, which is true as $\left( {10000 > 486} \right)$
Therefore from the above inequalities on substituting different values of n.
We can say that the inequality is true for $n \geqslant 5$
Option(C) is correct.

Note: Most of the students try to solve these types of inequalities using logarithm function or the exponential function it might give the correct answer but the process and the application are a bit complex to practice so we should always use the hit and trial method for questions like this.