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I'm trying to learn mathematical induction for proving inequalities, but there is just one step I cannot get past: finding another inequality that is added to the inductive hypothesis.

For example, in this problem:

Prove for all positive integers (n >= 1), prove 3^n + 2 >= 3n.

I understand the basis step and in general how to do induction, but for some reason, the example says that that after I get the hypothesis, 3^k + 2 >= 3k (for some arbitrary k), it can generate the inequality 2*3^k >= 3 for all k >= 1. Where does this come from? I can follow how it adds this inequality to the hypothesis, but what is this, and how would I go about getting this?

This isn't just a generic problem by the way: I've looked at many examples, but I can't figure out what this is when dealing with inequalities and induction.

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# Mathematical Induction Step

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