💯 The Inequality Proof with an Assumption by Math Induction Explained an Example

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1. Base case: Show that the statement is true for n = 2.
2^2 = 4, which is greater than 2 * 2 + 5 = 9. So the statement is true for n = 2.

2. Inductive step: Assume that the statement is true for some n = k, i.e. 2^k greater than or equal to 2k + 5. Show that it is also true for n = k + 1.
2^(k+1) = 2 * 2^k is greater than or equal to 2 * (2k + 5) = 2k + 2 * 5 = 2k + 10.
Since k is greater than 1, 2k + 10 is greater than 2 + 10 = 12, which is greater than or equal to 2 * (k + 1) + 5.
Hence, the statement is true for n = k + 1.

3. Conclusion: By the principle of mathematical induction, the statement is true for all positive integers n is greater than 1.