1. Mathematical induction

To prove a proposition related to a positive integer n, you can take the following steps:

(1) (inductive basis) proves that the proposition holds when n takes the first value n0 (n0 ∈ n+);

(2) The proposition of (inductive recursion) holds. If n = k (k ≥ n0, k ∈ n+), it is proved that the proposition also holds when n = k+ 1

As long as these two steps are completed, it can be concluded that the proposition holds for all positive integers n starting from n0.

2. Block diagram representation of mathematical induction

Prove the equation in the direction of 1 by mathematical induction

[Conventional method] 1. To prove the equation problem by mathematical induction, we should "look at the terms first" and find out the synthesis law of both sides of the equation, how many terms are there on each side of the equation and what is the initial value n0.

3. The proposition holds when n = k, and the equation holds when n = k+ 1 First of all, we should find out the changes (differences) on both sides of the equation and make clear the deformation target; Second, we should make full use of inductive hypothesis, make reasonable deformation and correctly write the proof process. Without proof of inductive hypothesis, it is not mathematical induction.

Test direction 2 prove inequality by mathematical induction

[Conventional method] 1. If it is not easy to prove the inequality related to positive integer n by other methods, mathematical induction can be considered.

2. The key to prove inequality by mathematical induction is that the proposition holds when n = k, and it also holds when n = k+ 1 After using inductive hypothesis, it can be proved by comparison, synthesis, analysis and scale, and the basic inequalities and the properties of inequalities can be fully applied to simplify the problem.

Test direction 3. Inductive conjecture proof

[Conventional method] 1. When guessing the general formula of {an}, we should pay attention to two points: (1) accurately calculate a 1, a2 and a3 to find the law (more items can be calculated if necessary); (2) When proving AK+ 1, the solution process of AK+ 1 is similar to a2 and a3, so we should pay attention to the dialectical relationship between the special and the general.

2. The model of "induction-conjecture-proof" is a comprehensive application of incomplete induction and mathematical induction. This method plays an important role in solving exploratory and existential problems. Its mode is to find the conclusion through reasonable reasoning first, and then prove the correctness of the conclusion through logical reasoning.

[Ideas and methods]

1. Mathematical induction is an important mathematical thinking method, which is mainly used to solve mathematical propositions related to positive integers. When proving, steps (1) and (2) are indispensable. Step (1) is the basis of step (2), which is the basis of recursion.

2. When deducing N = K+ 1, we can use inductive hypothesis by assembling, disassembling and matching articles. At this time, we should not only look at the target, but also understand the relationship between N = K and N = K+ 1. When deducing proof, we should flexibly use analysis, synthesis and reduction to absurdity.