The inequality 3 N> N 3holds when

Solving the Inequality $3^N > N^3$ for Natural Numbers The question asks for which natural numbers $N$ the inequality $3^N > N^3$ holds true. Natural numbers are $1, 2, 3, 4, \ldots$. To figure this out, let's test the inequality for the first few natural numbers and see what happens. Testing the Inequality for Small Natural Numbers We will compare the values of $3^N$ and $N^3$ for different natural number values of $N$. For $N=1$: $3^1 = 3$ $1^3 = 1$ Comparing, $3 > 1$. The inequality $3^N > N^3$ holds for $N=1$. For $N=2$: $3^2 = 9$ $2^3 = 8$ Comparing, $9 > 8$. The inequality $3^N > N^3$ holds for $N=2$. For $N=3$: $3^3 = 27$ $3^3 = 27$ Comparing, $27 > 27$ is false, because $27 = 27$. The inequality $3^N > N^3$ does NOT hold for $N=3$. For $N=4$: $3^4 = 81$ $4^3 = 64$ Comparing, $81 > 64$. The inequality $3^N > N^3$ holds for $N=4$. For $N=5$: $3^5 = 243$ $5^3 = 125$ Comparing, $243 > 125$. The inequality $3^N > N^3$ holds for $N=5$. Let's summarize the results in a table: $N$ $3^N$ $N^3$ $3^N > N^3$ ? 1 3 1 Yes 2 9 8 Yes 3 27 27 No ($27=27$) 4 81 64 Yes 5 243 125 Yes From the table, we can see that the inequality $3^N > N^3$ holds for $N=1$, $N=2$, $N=4$, $N=5$, and it seems likely it will continue to hold for $N > 3$. The only natural number among the first few tested for which the inequality does NOT hold is $N=3$. In fact, for $N \ge 4$, the exponential function $3^N$ grows much faster than the polynomial function $N^3$. We can try to prove this by induction or other methods, but for the purpose of checking the options, our initial tests are sufficient. Analyzing the Options Let's compare our findings with the given options: $N$ is any natural number: This is incorrect because the inequality does not hold for $N=3$. $N$ is a natural number greater than 2: This includes $N=3, 4, 5, \ldots$. We found it does not hold for $N=3$. So this option is incorrect. $N$ is a natural number greater than 3: This includes $N=4, 5, 6, \ldots$. While the inequality holds for these values, it also holds for $N=1$ and $N=2$, which are not greater than 3. This option is incomplete. $N$ is a natural number except 3: This means $N \in \{1, 2, 4, 5, 6, \ldots\}$. Our tests showed the inequality holds for $N=1, 2, 4, 5$. Since it holds for $N=1$ and $N=2$ and for all $N \ge 4$ (as exponential growth eventually surpasses polynomial growth), it holds for all natural numbers except $N=3$. This option matches our findings. Therefore, the inequality $3^N > N^3$ holds for all natural numbers $N$ except for $N=3$. Revision Table: $3^N$ vs $N^3$ Comparison for Natural Numbers Natural Number ($N$) Value of $3^N$ Value of $N^3$ Is $3^N > N^3$? 1 3 1 Yes 2 9 8 Yes 3 27 27 No 4 81 64 Yes 5 243 125 Yes ... ... ... Yes (for $N \ge 4$) Additional Information: Growth Rates of Functions This problem demonstrates the difference in growth rates between exponential functions and polynomial functions. An exponential function like $3^N$ grows much faster than a polynomial function like $N^3$ as $N$ becomes large. While for small values of $N$, a polynomial might be larger or equal (as seen with $N=3$), eventually, the exponential function will always overtake and stay larger than the polynomial function. This is a general principle in mathematics: exponential growth dominates polynomial growth for large inputs. In this specific case of $3^N$ versus $N^3$, the polynomial $N^3$ is greater than or equal to $3^N$ only for $N=3$. For all other natural numbers, $3^N$ is strictly greater than $N^3$.

Solve $3^N &gt; N^3$ Inequality for Natural Numbers

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CDS - 2019

Quantitative Aptitude

The inequality 3 ^N> N ³holds when

This question was previously asked in

CDS I 2019 Elementary Mathematics Previous Year Paper (03-Feb-2019)

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N is any natural number

N is a natural number greater than 2

N is a natural number greater than 3

N is a natural number except 3

The correct answer is

N is a natural number except 3

Solving the Inequality $3^N > N^3$ for Natural Numbers

The question asks for which natural numbers $N$ the inequality $3^N > N^3$ holds true. Natural numbers are $1, 2, 3, 4, \ldots$. To figure this out, let's test the inequality for the first few natural numbers and see what happens.

Testing the Inequality for Small Natural Numbers

We will compare the values of $3^N$ and $N^3$ for different natural number values of $N$.

For $N=1$: $3^1 = 3$ $1^3 = 1$ Comparing, $3 > 1$. The inequality $3^N > N^3$ holds for $N=1$.
For $N=2$: $3^2 = 9$ $2^3 = 8$ Comparing, $9 > 8$. The inequality $3^N > N^3$ holds for $N=2$.
For $N=3$: $3^3 = 27$ $3^3 = 27$ Comparing, $27 > 27$ is false, because $27 = 27$. The inequality $3^N > N^3$ does NOT hold for $N=3$.
For $N=4$: $3^4 = 81$ $4^3 = 64$ Comparing, $81 > 64$. The inequality $3^N > N^3$ holds for $N=4$.
For $N=5$: $3^5 = 243$ $5^3 = 125$ Comparing, $243 > 125$. The inequality $3^N > N^3$ holds for $N=5$.

Let's summarize the results in a table:

$N$	$3^N$	$N^3$	$3^N > N^3$ ?
1	3	1	Yes
2	9	8	Yes
3	27	27	No ($27=27$)
4	81	64	Yes
5	243	125	Yes

From the table, we can see that the inequality $3^N > N^3$ holds for $N=1$, $N=2$, $N=4$, $N=5$, and it seems likely it will continue to hold for $N > 3$. The only natural number among the first few tested for which the inequality does NOT hold is $N=3$.

In fact, for $N \ge 4$, the exponential function $3^N$ grows much faster than the polynomial function $N^3$. We can try to prove this by induction or other methods, but for the purpose of checking the options, our initial tests are sufficient.

Analyzing the Options

Let's compare our findings with the given options:

$N$ is any natural number: This is incorrect because the inequality does not hold for $N=3$.
$N$ is a natural number greater than 2: This includes $N=3, 4, 5, \ldots$. We found it does not hold for $N=3$. So this option is incorrect.
$N$ is a natural number greater than 3: This includes $N=4, 5, 6, \ldots$. While the inequality holds for these values, it also holds for $N=1$ and $N=2$, which are not greater than 3. This option is incomplete.
$N$ is a natural number except 3: This means $N \in \{1, 2, 4, 5, 6, \ldots\}$. Our tests showed the inequality holds for $N=1, 2, 4, 5$. Since it holds for $N=1$ and $N=2$ and for all $N \ge 4$ (as exponential growth eventually surpasses polynomial growth), it holds for all natural numbers except $N=3$. This option matches our findings.

Therefore, the inequality $3^N > N^3$ holds for all natural numbers $N$ except for $N=3$.

Revision Table: $3^N$ vs $N^3$ Comparison for Natural Numbers

Natural Number ($N$)	Value of $3^N$	Value of $N^3$	Is $3^N > N^3$?
1	3	1	Yes
2	9	8	Yes
3	27	27	No
4	81	64	Yes
5	243	125	Yes
...	...	...	Yes (for $N \ge 4$)

Additional Information: Growth Rates of Functions

This problem demonstrates the difference in growth rates between exponential functions and polynomial functions. An exponential function like $3^N$ grows much faster than a polynomial function like $N^3$ as $N$ becomes large. While for small values of $N$, a polynomial might be larger or equal (as seen with $N=3$), eventually, the exponential function will always overtake and stay larger than the polynomial function. This is a general principle in mathematics: exponential growth dominates polynomial growth for large inputs.

In this specific case of $3^N$ versus $N^3$, the polynomial $N^3$ is greater than or equal to $3^N$ only for $N=3$. For all other natural numbers, $3^N$ is strictly greater than $N^3$.

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