Final Answer:
Mathematical induction can be proven by verifying the base case, formulating an inductive hypothesis, recursively taking steps, and generalizing to all cases, thus the correct option is B.
Explanation:
Mathematical induction is a proof technique used to prove that a statement holds for all natural numbers. It works by assuming that a statement is true for some natural number (the base case) and then proving that if it holds for any natural number, it holds for the next one (the inductive step). This process is repeated until the statement is proven to hold for all natural numbers.
To prove a given statement using mathematical induction, one must first verify the base case. This is done by checking to make sure that the statement holds when the natural number is equal to the base case. Once this is established, an inductive hypothesis must be formulated. This is done by assuming the statement holds for any natural number n, and then proving that it holds for n+1. After this, the recursive step is taken. This involves taking the inductive hypothesis, using it to prove the statement holds for n+1, and then repeating the process for n+2, n+3, etc. until the statement holds for all natural numbers. Finally, the generalization to all cases is made. This involves taking the recursive step and using it to prove the statement holds for all natural numbers, thus completing the proof.
In conclusion, mathematical induction can be proven by verifying the base case, formulating an inductive hypothesis, recursively taking steps, and generalizing to all cases. This method is useful for proving statements that hold for all natural numbers and can be used to prove theorems in number theory and other areas of mathematics.