Final Answer:
(a) Using mathematical induction, we can prove that P(n) is true for all integers n ≥ 12. (b) Employing strong induction, we can also establish the truth of P(n) for all integers n ≥ 12.
Step-by-step explanation:
In part (a), we use mathematical induction to prove that the statement P(n) holds for all integers n ≥ 12. The base case, n = 12, is checked to ensure P(12) is true. Then, assuming P(k) is true for some arbitrary k, we prove P(k+1). This establishes the induction step, confirming P(n) for all n ≥ 12.
In part (b), strong induction is applied. Similar to mathematical induction, the base case is verified for n = 12. However, in the induction step, instead of assuming P(k), we assume P(j) is true for all j where 12 ≤ j ≤ k. This stronger assumption allows for a more comprehensive induction step, reinforcing the proof that P(n) is true for all n ≥ 12.
Comparing the two proofs in part (c), mathematical induction relies on the assumption that P(k) is true, while strong induction assumes the truth of P(j) for all j between the base case and k. Strong induction can be seen as a more flexible approach, as it considers a broader range of assumptions in the induction step.
However, in some cases, mathematical induction may be simpler to apply. Both methods are valid and lead to the same conclusion, but the choice between them depends on the nature of the problem at hand.