Mathematical Induction

Mathematical induction is a powerful technique used to prove statements about natural numbers. It is a method of proving that an assertion holds for all integers greater than or equal to a particular starting value. The process of mathematical induction involves two steps:

1. Base case: Prove that the statement holds for the starting value, usually 0 or 1.
2. Induction step: Prove that if the statement holds for some integer n, then it also holds for the next integer n+1.

By combining these two steps, we can prove that the statement holds for all natural numbers greater than or equal to the starting value.

Example

As an example, let's prove that the sum of the first n natural numbers is given by the formula $\frac{n(n+1)}{2}$ for all natural numbers n.

Base case: When n = 1, the sum of the first natural number is 1, which is equal to $\frac{1(1+1)}{2}$. Therefore, the formula holds for n = 1.

Induction step: Assume that the formula holds for some arbitrary natural number k. That is, the sum of the first k natural numbers is given by $\frac{k(k+1)}{2}$. Now we need to prove that the formula also holds for k+1.

By definition, the sum of the first k+1 natural numbers is the sum of the first k natural numbers plus the next number, which is k+1. Thus, we can write:

$1 + 2 + 3 + ... + k + (k+1) = \frac{k(k+1)}{2} + (k+1)$

Simplifying the right-hand side of the equation yields:

$\frac{k(k+1)}{2} + (k+1) = \frac{(k+1)(k+2)}{2}$

Thus, we have shown that the formula holds for k+1 as well. Therefore, by mathematical induction, the formula holds for all natural numbers n.

Conclusion

Mathematical induction is a powerful technique for proving statements about natural numbers. It involves proving a base case and an induction step to show that the statement holds for all natural numbers greater than or equal to a starting value. By breaking down the proof into these simple steps, we can prove complex statements about natural numbers with ease.

Links

数学的帰納法[JA]