Wiskundige inductie

Mathematical induction is a technique used to establish the truth of a statement for all natural numbers. For example, in a domino effect, if the first one falls and each one topples the next, the entire line will fall.

Consider another example of saving quarters daily—1 quarter on the first day, 2 on the second, 3 on the third, and so on, up to n days. The total number of quarters forms a series that includes the first n natural numbers. The sum of this series follows a simple pattern that, when evaluated, is equal to n times n plus 1 over 2.

Mathematical induction proves this pattern for all natural numbers by confirming the base case and the inductive step.

For the first number, the base case, the actual sum matches the result from the rule, which verifies the base case.

The assumption is then made that the rule works for any number k. This forms the inductive step, where the sum up to that number is expected to follow the same pattern.

Adding the next number, k plus 1, is shown to maintain the pattern, following the same logic.

With both the base case and the inductive step holding true, the rule is validated for all natural numbers through mathematical induction.