Chapter 6: Problem 479

If \(\mathrm{P}(\mathrm{n}): 3^{\mathrm{n}}<\mathrm{n} !, \mathrm{n} \in \mathrm{N}\), then \(\mathrm{P}(\mathrm{n})\) is true (a) for \(\mathrm{n} \geq 6\) (b) for \(n \geq 7, n \in N\) (c) for \(n \geq 3\) (d) for all \(n\)

Short Answer

Expert verified

The inequality \(3^n < n!\) holds true for \(n \geq 7\) where n belongs to natural numbers. Therefore, the correct choice is (b) for \(n \geq 7, n \in N\).

Step by step solution

Determine the Range

To find the range of n for which the given inequality is true, let's compare the values of \(3^n\) and \(n!\) for different natural number values of n. \(n \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ }\) \(3^n \textrm{ }\) \(n!\) \(1 \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ } 3 \textrm{ } \textrm{ }\) \(1\) \(2 \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ } 9 \textrm{ }\) \(2\) \(3 \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ } 27\)\(6\) \(4 \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ } 81\)\(24\) \(5 \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ } 243\)\(120\) \(6 \textrm{ } \textrm{ } \textrm{ } \textrm{ } \textrm{ } 729\)\(720\) \(7 \textrm{ } \textrm{ } \textrm{ } \textrm{ } 2187\)\(5040\) As we can observe from the above table, the inequality \(3^n < n!\) holds true for \(n \geq 7\).

Compare the Range with Answer Choices

Based on our findings, the inequality \(3^n < n!\) holds true for \(n \geq 7\) where n belongs to natural numbers. Looking at the given options, the correct choice is (b) for \(n \geq 7, n \in N\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorial

The factorial of a natural number is the product of all positive integers up to that number. For example, the factorial of 3, denoted as 3!, is calculated as 3 × 2 × 1, which equals 6. The concept of factorial grows very rapidly with larger numbers. This is due to the multiplying effect of each sequential natural number; the bigger the number, the larger the factorial becomes, far outpacing linear or even exponential growth in comparison.

Factorials are fundamental in various mathematical fields, including permutations, combinations, and in the calculation of probabilities. In the provided exercise, the factorial represents the growing complexity of arrangements as the number set increases, which is why beyond a certain point, it becomes significantly larger than the corresponding exponential term of the same number.

Exponential Growth

Exponential growth is a process that increases quantity over time. It occurs when the growth rate of a mathematical function is proportional to the current value, leading to growth that occurs at a progressively increasing rate. For example, in the formula 3^n, the number 3 is raised to the power of n. As n increases, the value of 3^n grows exponentially.

The Nature of Exponential Functions

In these functions, each increment of the exponent results in the value of the base number being multiplied by itself. This property can lead to very large numbers quickly, but, as it turns out, not as quickly as factorial growth for larger values of n. In our exercise, we compare this exponential growth to factorial growth to determine at which point the factorial overtakes the exponential expression.

Natural Numbers

Natural numbers are the set of positive integers that we use for counting and ordering. In mathematics, they are usually denoted by N and include all the positive integers from 1 onwards: 1, 2, 3, 4, 5, and so on. Zero is not considered a natural number when considering factorials.

Natural numbers have a fundamental role in mathematics and are especially important when defining factorials, as factorials only apply to these numbers. The operation of factorial is not defined for negative numbers or fractions. In the context of the given exercise, we limit our study of inequalities to natural numbers because we're dealing with factorials, which are inherently a concept tied to the set N.

Mathematical Induction

Mathematical induction is a powerful proof technique used in mathematics to prove an infinite series of statements or properties. It works in a two-step process: first, showing that the statement is true for the first natural number in the series (base case), and then, assuming it is true for some arbitrary natural number 'k,' proving that it must also be true for the next number 'k+1' (inductive step).

This method is often used to establish that a given property P(n) holds for all numbers n greater than or equal to a certain number. In the context of our factorial and exponential growth comparison, induction can be used to show that once the inequality holds for a particular number, it continues to hold for all larger natural numbers, as both the factorial and the exponentiation scale up.

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If \(\mathrm{P}(\mathrm{n}): 3^{\mathrm{n}}<\mathrm{n} !, \mathrm{n} \in \mathrm{N}\), then \(\mathrm{P}(\mathrm{n})\) is true (a) for \(\mathrm{n} \geq 6\) (b) for \(n \geq 7, n \in N\) (c) for \(n \geq 3\) (d) for all \(n\)

Short Answer

Step by step solution

Determine the Range

Compare the Range with Answer Choices

Key Concepts

Factorial

Exponential Growth

The Nature of Exponential Functions

Natural Numbers

Mathematical Induction

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