Question

Use mathematical induction to prove that \(2^{n}

Short answer

Question: Use mathematical induction to prove the inequality \(2^{n}<n!\) for all integers \(n \geq 4\). Answer: The inequality \(2^{n}<n!\) has been proven to hold for all integers \(n \geq 4\) using mathematical induction. The base case (\(n=4\)) was tested, and the inequality held true. Then, we assumed the inequality held for an arbitrary integer \(k \geq 4\) and successfully proved it held for \(k+1\). By the principle of mathematical induction, the inequality \(2^n < n!\) is valid for all integers \(n \geq 4\).

Step-by-step solution

Test the base case

For \(n = 4\), we need to test if \(2^n < n!\). Plugging in the values, we get \(2^4 < 4!\), which simplifies to \(16 < 24\). This inequality holds, thus verifying the base case.

Inductive Hypothesis and Inductive Step

Assume that, for an arbitrary integer \(k \geq 4\), the inequality \(2^{k} < k!\) is true. We now need to prove that the inequality holds for \(k+1\): \(2^{k+1}<(k+1)!\).

Prove the inequality for \(k+1\) using the Inductive Hypothesis

To prove the inequality \(2^{k+1}<(k+1)!\), we can start by multiplying both sides of the assumed inequality \(2^{k} < k!\) by 2: \(2(2^{k}) < 2(k!)\). Note that \(2(2^{k}) = 2^{k+1}\). Next, we need to show that \(2(k!) < (k+1)!\). Recall that \((k+1)! = (k+1)k!\). Since we have assumed \(k \geq 4\), clearly \(k+1 > 2\). This means that \((k+1)k! > 2(k!)\), which implies \(2^{k+1}<(k+1)!\).

Conclusion

Since we have successfully proven that the inequality \(2^{n} < n!\) holds for the base case \(n = 4\) and for \(k+1\) when assuming it is true for an arbitrary integer \(k \geq 4\), we can conclude that by the principle of mathematical induction, the inequality \(2^n < n!\) is valid for all integers \(n \geq 4\).

Key concepts

Proof by Induction

Understanding the concept of proof by induction is essential when exploring mathematical truths that recur or build upon themselves. It is a form of direct proof used primarily for proving statements or the validity of equations for all natural numbers. Mathematical induction involves two critical steps: verifying the base case and conducting the inductive step.

For the base case, choose the smallest integer for which the statement should hold and prove it is true. In our exercise, the base case is when n equals 4. If the base case is true, we move to the inductive step, where we assume that the statement is true for some arbitrary natural number k (the inductive hypothesis). Then we prove that if the statement holds for k, it must hold for k+1. This step often requires algebraic manipulation and using the inductive hypothesis effectively. Upon proving the inductive step, we conclude that the statement is true for all integers greater than or equal to our base case, thanks to the domino effect.

The key is to ensure clear, logical reasoning through each phase, so the proof's foundational logic is irrefutable—a process similar to knocking down an endlessly lined-up set of dominoes, where knocking down the first ensures all subsequent ones follow.

Factorial

The factorial function (!) has a simple definition but plays a profound role in combinatorics, probability, and various mathematical fields. A numeral's factorial is the product of all positive integers less than or equal to that number. For instance, for an integer n, the factorial n! is calculated as n × (n-1) × (n-2) ... × 3 × 2 × 1.

For our exercise, understanding factorials allows us to compare the growth of factorial expressions to exponential functions. Factorials grow at an extremely fast rate, much faster than exponential functions, especially as the numbers increase. This property comes into play when distinguishing between inequalities involving factorials and exponentials. It's vital for students to grasp that as n grows larger, the difference between and n! becomes vast and far-reaching.

Inequalities

Inequalities are mathematical expressions that describe the relative size or order of two values. They differ from equations in that they do not show equality, instead utilizing symbols to express which value is larger or smaller than the other. Common symbols include ' > ' (greater than), ' < ' (less than), ' ! It's essential not to get 'intimidated' by inequalities – approach them step-by-step, and remember that manipulating an inequality almost always requires consideration of the direction of the inequality and the properties of the numbers involved, like whether they are positive or negative.

Exponential Functions

Exponential functions are characterized by their variable exponent; the base is a constant, and the power is a variable, expressed as f(x)=a^x, where 'a' is a constant and 'x' is the variable. These functions are crucial in describing phenomena with constant growth rates, like population growth or radioactive decay.

When we look at the solution to our exercise, it is clear that exponential functions, such as 2^n, increase rapidly but not fast enough to outpace a factorial's growth, such as n!. A solid grasp of how exponential functions operate significantly aids in understanding their comparison to other forms of advanced growth. Remember, exponential functions might start off growing slower than linear or quadratic functions (when n is small), but as n becomes large, their rate of growth rapidly overtakes and exceeds almost any polynomial function.