Discover the law of solving problems;
First, observation.
Observation is the most basic and direct method to discover the law. By observing the characteristics and laws of series, figures and equations, we can try to find the laws. For example, observing the law of the difference between the first few items of a series, the law of product, the relationship between adjacent items, etc., can help us find a general term formula of a series.
Second, algebraic method.
Algebraic method is a method to find the law through algebraic operation. By establishing a mathematical model, the problem is represented by algebraic symbols, and then the solution of the problem is obtained by derivation and calculation with algebraic knowledge. Algebraic methods are usually suitable for solving some complex problems, such as equations and inequalities.
Third, induction.
Induction is a method to sum up some known results and deduce the general situation. By observing a series of examples or special situations, we can sum up the law and prove that this law is applicable to all situations. Induction is often used to prove mathematical theorems and solve some complicated problems.
Fourthly, recursive method.
Recursive method is a method to deduce the next item from the known item through the known conditions and recursive relations. Recursive methods are often used to solve problems such as series and tables. By looking for the relationship between adjacent items in a series or table, we can infer the following items.
Five, the combination of number and shape method
The combination of numbers and shapes is a method of combining mathematics with geometric figures to find laws. The problem can be solved by transforming mathematical problems into geometric problems, or by drawing and drawing. The combination of numbers and shapes is often used to solve some geometric and graphic problems.
Sixth, reduce to absurdity.
Reduction to absurdity is a method to prove the original proposition by assuming the opposite side of the problem and then deducing a conclusion that is contradictory to what is known. In the process of finding the law, we can assume that a certain law is established, and then verify whether this law is correct by reducing to absurdity.
Seven, mathematical induction
Mathematical induction is a common method to prove mathematical propositions. By first proving that the proposition holds in a certain situation, then assuming that the proposition holds in a certain situation, and then proving that the proposition also holds in the next situation, we finally come to the conclusion that the proposition holds in all cases.
Eight, analysis method
Analytic method is a method to decompose a problem into thousands of sub-problems and then solve them one by one. Through the analysis of the problem, we can better understand the structure and characteristics of the problem and find the law of the problem.