nastygoalie89 said:Homework Statement
5^n + 9 < 6^n for all integers n>=2.
Homework Equations
The Attempt at a Solution
Induction proof.
Base Case: 5^(2) + 9 < 6^(2)
34<36
P(k): 5^k + 9 < 6^k
P(k+1): 5^(k+1) + 9 < 6^(k+1)
how do i prove p(k) can equal p(k+1)?
The purpose of proving this inequality is to demonstrate the relationship between the powers of 5 and 6 for values of n greater than or equal to 2. This can help to understand the growth rate of these exponential functions and make predictions about their values for larger values of n.
The process for proving this inequality is through mathematical induction. This involves showing that the inequality holds true for a base case (in this case, n=2), and then assuming it holds true for a general case (n=k) and using this assumption to prove that it also holds true for the next case (n=k+1). This process continues until the inequality is proven to hold true for all values of n greater than or equal to the base case.
Mathematical induction is a commonly used proof technique in mathematics, particularly for proving statements that involve patterns or sequences. In this case, the inequality involves powers of numbers, which follow a certain pattern. Using mathematical induction allows for a systematic way to prove that the inequality holds true for all values of n, rather than having to test each individual case.
Yes, it is possible to prove this inequality using other methods such as direct proof or contradiction. However, mathematical induction is the most commonly used method for proving inequalities involving patterns or sequences, as it provides a clear and organized approach.
Proving this inequality can have various applications in mathematics and other fields. It can be used to understand the growth rate of exponential functions, make predictions about their values, and solve related problems. It can also be used as a building block for proving other inequalities and mathematical statements.