(1) inductive basis: prove that the proposition holds when n= 1;
(2) Inductive hypothesis: the proposition holds when n=k is assumed;
(3) Inductive recursion: The proposition also holds when n=k+ 1 is deduced from inductive hypothesis.
The second principle of mathematical induction is that there is a proposition related to the natural number n, if:
(1) When n = 1, the proposition holds;
(2) Assuming that the proposition holds when n≤k, it can be inferred that the proposition also holds when n = k+ 1.
Then, this proposition holds for all natural numbers n.
Extended data:
In number theory, mathematical induction is a mathematical theorem that proves that any given situation is correct in different ways (the first, the second and the third, all the way down, no exception).
Although there is "induction" in the name of mathematical induction, mathematical induction is not a rigorous inductive reasoning method, but a completely rigorous deductive reasoning method. In fact, all mathematical proofs are deductive.
Mathematical induction has strict requirements on the form of solving problems. In the process of solving problems by mathematical induction,
Step 1: Verify that n holds when it takes the first natural number.
Step 2: Assume that n=k holds, and then deduce it according to the conditions of verification and hypothesis. In the following derivation process, n=k+ 1 cannot be directly substituted into the assumed original formula.
The last step is to summarize the statement.
It should be emphasized that the two steps of mathematical induction are both very important and indispensable.
The principle of mathematical induction is usually defined as the axiom of natural numbers (see Piano's axiom). But on the basis of other axioms, it can be proved by some logical methods. The principle of mathematical induction can be derived from the following axiom of good order (principle of minimum natural number):
Natural number set is orderly. (Every non-empty positive integer set has a minimum element)
For example, the number of {1, 2,3,4,5} is the smallest-1.
Below we will prove mathematical induction through this property:
For the mathematical proposition that has been proved in the above two steps, we assume that it is not true for all positive integers.
There must be a minimum element k in the set s composed of those invalid numbers. (1 does not belong to the set s, so k >: 1)
K is already the smallest element in the set S, so k- 1 does not belong to S, which means that k- 1 is valid for the proposition-since it is valid for k- 1, it should also be valid for K, which contradicts the second step we completed. So this two-step proposition holds for all n.
It is worth noting that some other axioms are actually alternative axioms of mathematical induction principle. More precisely, the two are equivalent.
References:
Baidu Encyclopedia-Mathematical Induction