Get the intuition behind proof by induction

Question

Any proof which is proven by direct proof method is easy to understand and feel. How do you get the feel or intuition behind the proof when a theorem or lemma is proved using the principle of mathematical induction?

Answer 1

If you are talking intuition:

In a proof by induction we prove two things.

2) If something is true for one value, it will always have to be true for the next value.

1) Something it true for the first value.

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To me it is very intuitive that that means something is true for the first value (by step 1), so it is true for the second value (by step 2), so it is true for the third value (by step 2 again), so it is true by the fourth value (by step 2 yet again), and therefore by repeating step 2) any number of times, it is true for any value we can count to.

And intuitively, we can (given enough time) count to any positive whole number (supposedly). So give those two facts, it intuitively follows the something is true for all values.