Final answer:
In an induction proof, to show that P(k+1) is true, we assume P(k) is true and then demonstrate that P(k) being true implies that P(k+1) must also be true, thus confirming the pattern for all positive integers.
Step-by-step explanation:
Inductive Proof in Mathematics
To show that P(k+1) is true during an induction proof, we assume that P(k) is true for some arbitrary positive integer k—this is known as the induction hypothesis. We then use this hypothesis to demonstrate that the statement holds for the next case, P(k+1). This process involves thorough reasoning and the application of mathematical operations that stem from the assumption that P(k) is valid to show that it logically implies the truth of P(k+1). If we can show that P(1) is true (the base case) and that if P(k) is true then P(k+1) is also true, we can conclude by induction that P(n) is true for all positive integers n. This method of reasoning allows us to make generalizations or patterns about infinite sets based on finite cases.