Mathematical induction is a proof method, based on the increment of natural numbers. By proving that the proposition holds on the basis of the previous natural number, it also holds on the next natural number.

Mathematical induction generally includes two steps:

1. Basic steps: The proof proposition is based on an initial natural number, and usually the proof proposition is based on a natural number of 1.

2. Inductive step: suppose that the proposition holds on the natural number n, and then prove that the proposition also holds on the natural number n+ 1.

Through the repeated application of these two steps, mathematical induction can infer that this proposition holds true for all natural numbers.

Because natural numbers are infinite and have the nature of increasing, mathematical induction can be well applied to prove propositions related to natural numbers, such as:

Mathematical induction can prove the summation formula of any natural number n:1+2+3+...+n = n (n+1)/2.

Mathematical induction can prove arithmetic progression's general formula: an = a1+(n-1) d.

Mathematical induction can prove the recurrence relation of Fibonacci sequence: F(n)=F(n- 1)+F(n-2).

Mathematical induction is a method commonly used to prove mathematical propositions. Its basic idea is to prove that when a proposition holds in a specific case, and assume that it holds in a specific integer n, then prove that it also holds in the case of n+ 1 In this way, we can deduce that the proposition holds for all integers greater than or equal to the initial situation.

complete induction

First of all, let's set a proposition P(n), and we have to prove that it holds for all integers n greater than or equal to an integer n0. Let's first prove that P(n0) is effective, which is the basic step.

Next, we assume that P(n) holds for an integer n, and use this assumption to prove P(n+ 1).

In the induction step, we use mathematical operations and logical reasoning to prove P(n+ 1).

Usually, we need to bring P(n+ 1) into the expression of P(n) to simplify the calculation, and finally get the conclusion that P(n+ 1) holds.

Finally, through the basic steps and inductive steps, it is concluded that the proposition P(n) is valid for all integers n greater than or equal to n0.

Mathematical induction is widely used in mathematical proof. It can be used to prove various propositions, such as equality, inequality, combinatorial formula, the nature of sequence and so on. Using mathematical induction, we can draw conclusions step by step and establish the logic and rigor of mathematical thinking.

In the proof of mathematical induction, we need to ensure the correctness of the basic steps and induction steps. The basic step is the smallest case, which can usually be verified by examples. The induction step requires logical reasoning and mathematical operation to ensure that the process from P(n) to P(n+ 1) is correct.