Mathematical induction is just a way of proving a statement to be true for all positive integers: prove the statement to be true about 1; then assume it to be true for a generic integer x, and prove it to be true for x + 1; it therefore must be true for all positive integers.
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No, the problem of induction is too circular to be solved. Read some Thomas s. Kuhn or Karl Popper.
contradiction
I am not really sure what you are asking but there are 3 types of proofs in geometry a flow proof, a 2-collumn proof, and a paragraph proof.
Neither. It is an IDENTITY.
A proof that uses techniques from geometry.