💯 How to Find the Initial Value of an Inequality Proof by Math Induction Explained

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1. Base case: Show that the statement is true for n = 1.
1^2 + 1 = 2, which is less than 1^3 - 1 = 0. So the statement is true for n = 1.

2. Inductive step: Assume that the statement is true for some n = k, i.e. k^2 + k is less than k^3 - 1. Show that it is also true for n = k + 1.
(k+1)^2 + (k+1) = k^2 + 2k + 1 + k + 1 = k^2 + k + 2k + 2 = k^3 + 2(k + 1) = k^3 + 2k + 2.
k^3 - 1 is less than k^3 + 2k + 2 = (k + 1)^2 + (k + 1) for positive k.
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.