mathematical induction

What is mathematical induction?

Mathematical induction is a proof technique used to prove that a statement P(n) is true for every natural number n. It is particularly useful for proving properties of sequences, sums, or algorithms that involve natural numbers.

What are the two main steps in a proof by mathematical induction?

The two main steps are:

1. Base Case (or Basis Step): Show that the statement P(1) is true.
2. Inductive Step: Show that if P(k) is true for some arbitrary natural number k, then P(k+1) is also true. This is often stated as P(k) \implies P(k+1).

Explain the Base Case in mathematical induction.

The Base Case (or Basis Step) involves proving that the statement P(n) holds for the smallest value of n in the domain of interest, often n=1 (or n=0 or some other integer).

This step establishes the 'starting point' for the inductive 'chain reaction'.

Explain the Inductive Hypothesis in mathematical induction.

The Inductive Hypothesis is the assumption made during the Inductive Step that the statement P(k) is true for some arbitrary natural number k \geq 1 (given that the base case starts at n=1).

This assumption is crucial for proving that P(k+1) is also true.

Explain the Inductive Step in mathematical induction.

The Inductive Step involves proving that if the Inductive Hypothesis (P(k) is true) holds, then the statement P(k+1) must also be true.

It demonstrates that the truth of the statement 'propagates' from one natural number to the next, similar to how falling dominoes knock over the next one.