Question

In the induction proofs how do we use P(k) and P(k+1)?

Answer 1

In induction proofs, P(k) denotes a proposition for a natural number k, assumed true in the induction hypothesis. The goal is to show that if P(k) is true, P(k+1), the proposition for the next number, is also true, completing the induction step. Through algebraic manipulation, we demonstrate P(k+1) based on the assumption of P(k).

Step-by-step explanation:

In induction proofs, we use P(k) to denote a proposition that is assumed to be true for a given natural number k. This is known as the induction hypothesis. The goal of the proof is to show that if P(k) is true, then P(k+1) must also be true, which is the proposition for the next natural number, k+1. This step is called the induction step.

To perform the induction step effectively, we typically:

1. Assume P(k) is true (the induction hypothesis).
2. Based on that assumption, manipulate the expression or inequality that represents P(k) to create a form that resembles P(k+1), often through algebraic simplification, adding a term, or factoring.
3. Show that these manipulations lead logically to the truth of P(k+1), thereby proving the original proposition for all natural numbers, assuming the base case has already been proven.

Induction proofs are essential in proving propositions that are hypothesized to be true for all natural numbers or for numbers within a certain range.