Introduction
Mathematical induction is a powerful technique used to prove statements about natural numbers. In this article, we will demonstrate how to prove the inequality ( n! geq 3^n ) for ( n geq 7 ) using the principle of mathematical induction. We will break down each step, from the base case to the inductive step, to ensure a thorough understanding of the process.
Base Case
To establish the base case, we check if the statement holds for ( n 7 ).
Calculate ( 7! ):
[7! 7 times 6 times 5 times 4 times 3 times 2 times 1 5040]Calculate ( 3^7 ):
[3^7 3 times 3 times 3 times 3 times 3 times 3 times 3 2187]Clearly, ( 5040 geq 2187 ), so the base case is satisfied.
Inductive Step
Assume the statement is true for some ( n k ), where ( k geq 7 ). Therefore, we assume:
[k! geq 3^k]We need to demonstrate that this implies:
[(k 1)! geq 3^{k 1}]Expressing ((k 1)!)
Express ((k 1)!) as:
[(k 1)! (k 1) cdot k!]Using the Inductive Hypothesis
Using our inductive hypothesis ( k! geq 3^k ), we get:
[(k 1)! (k 1) cdot k! geq (k 1) cdot 3^k]Showing ((k 1) cdot 3^k geq 3^{k 1})
Now, we need to show:
[(k 1) cdot 3^k geq 3^{k 1}]This simplifies to:
[(k 1) cdot 3^k geq 3 cdot 3^k]Divide both sides by ( 3^k ), which is positive for ( k geq 7 ):
[k 1 geq 3]This inequality is true for ( k geq 2 ). Since our induction starts at ( k 7 ), this condition is satisfied.
Conclusion
Since both the base case and the inductive step have been verified, we conclude that by mathematical induction:
[n! geq 3^n quad text{for all } n geq 7.]This rigorous proof demonstrates that the factorial of a number is always greater than or equal to ( 3^n ) for ( n ) starting at 7.
Additional Notes
Note that for ( n 6 ), ( 6! 720 ) and ( 3^6 729 ), which does not satisfy the inequality. However, once ( n geq 7 ), the inequality holds true as demonstrated by the base case and the inductive step.
Mathematical induction is a crucial tool in proving statements about sequences and series. The method of induction relies on the principle that if a statement is true for the first case, and the truth of the next case can be established from the previous case, then the statement is true for all subsequent cases.