10.5
Mathematical Induction
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.
Mathematical induction is a structured method of proof used to confirm the truth of statements involving natural numbers. Consider the sum of the first n natural numbers:
This formula describes a pattern that appears to hold true as more terms are added. To verify that it is valid for all natural numbers, mathematical induction proceeds in two essential steps. The first is the base case, where the formula is tested for the initial value, typically n = 1. Substituting into both sides confirms the equality:
so the base case holds. Next is the inductive step, where the formula is considered true for some arbitrary natural number k. This assumption is known as the induction hypothesis:
Then, it is shown that the formula also holds for k + 1. Adding k + 1 to both sides of the induction hypothesis gives:
This confirms the pattern continues for k + 1. With the base case and inductive step verified, the formula is proven for all natural numbers.
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.
From Chapter 10:
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