Mathematical induction

Japanese school year: Math B

What you learn

Learn how to use mathematical induction, a powerful proof technique for statements involving natural numbers. It is widely used to verify summation formulas, divisibility properties, and sequence behaviors. A solid grasp of algebraic manipulation and basic logical deduction will prepare you well for this concept.

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Key points

To prove a proposition for all natural numbers, first verify it holds for $n = 1$ , then prove that if it holds for $n = k$ , it must also hold for $n = k + 1$ .

If the statement is true for $n = 1$ and each step passes truth to the next like falling dominoes, $P(n)$ is proven true for all natural numbers $n$ .

[P(1) \land \forall k (P(k) \implies P(k+1))] \implies \forall n P(n)

When proving the case for $n = k + 1$ , the key is to rewrite the expression so that you can substitute in the formula assumed for $n = k$ .

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Practice 1Coming soon