Math Problem Statement
Solution
The image shows a mathematical inequality that needs to be proven using mathematical induction. The inequality is:
This can be proven using the principle of mathematical induction. Here's a step-by-step outline of how the induction proof would go:
Step 1: Base Case
First, check if the statement holds for .
Clearly, , so the base case holds.
Step 2: Inductive Hypothesis
Assume the statement is true for some , that is, assume:
Step 3: Inductive Step
We now need to prove that the inequality holds for , i.e., prove that:
Using the inductive hypothesis, we know that . Now, multiply both sides of by 2:
So, we need to show that . Expanding both sides:
Thus, we need to prove:
This simplifies to:
This inequality holds for , as we can verify by plugging in values of .
Conclusion:
Since the base case holds and the inductive step is proven, by mathematical induction, the inequality holds for all .
Do you have any questions or would you like more details on any of these steps?
Here are 5 related questions you might find useful:
- How do you apply mathematical induction to prove inequalities?
- Why is the base case important in an induction proof?
- Can mathematical induction be used for non-integer values of ?
- What happens if the base case fails in an induction proof?
- How do you simplify expressions involving powers in induction proofs?
Tip: In mathematical induction, always verify the base case carefully, as it sets the foundation for the entire proof.
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Math Problem Analysis
Mathematical Concepts
Mathematical Induction
Exponential Growth vs Quadratic Growth
Formulas
2^n ≥ n^2 for n ≥ 9
Theorems
Principle of Mathematical Induction
Suitable Grade Level
Undergraduate (or advanced high school math)