Math Problem Statement
since we already have the base case as n = 0, we can assume k > 0 for our inductive hypothesis.
Solution
To prove the inequality for all integers using mathematical induction, we can follow these steps:
Step 1: Base Case
We check the base case :
The inequality holds true for .
Step 2: Inductive Hypothesis
Assume that the inequality holds for some integer . This is our inductive hypothesis.
Step 3: Inductive Step
We need to show that if , then as well.
Starting with the left side, we use the property of exponents to express :
Since we assume , we can multiply both sides of this inequality by 2:
Thus,
Now, we need to show that , which would imply .
Rearrange to . Therefore, for ( k \geq 1 \
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Math Problem Analysis
Mathematical Concepts
Mathematical Induction
Inequalities
Exponents
Formulas
2^n > n
Theorems
Principle of Mathematical Induction
Suitable Grade Level
Grades 10-12