There are two main steps and a conclusion:
(1) proves that the conclusion is correct when n takes the first value n0 (such as n0= 1 or 2, etc.). ).
(2) Assuming that n = k (k ∈ n+, and k≥ n0), the conclusion is correct.
It is also proved that the conclusion is correct when n=k+ 1.
The conclusions of (1) and (2) are correct (the proposition holds).
Note: 1. N0 is not necessarily equal to 1.
2. The number of projects does not necessarily increase by only one project.
3. Be sure to use assumptions (note: use assumptions
Recursion is true)
Steps and points for attention in proving identities by mathematical induction;
① Make clear the initial value n0 and verify the authenticity. (basic)
(2) "Assuming that the proposition is correct when n=k", write the proposition form.
③ Analyze the proposition "when n=k+ 1" and find out the difference from "when n=k".
The difference in the form of proposition, find out the item that should be added at the left end.
(4) Define the deformation target at the left end of the equation and master the common deformation of the identity.
Methods: Multiplication formula, factorization, addition, subtraction, multiplication and division, formula, etc.
And be sure to use assumptions.
The basic form of mathematical induction
1. The first mathematical induction
Let be a proposition related to positive integers, if
(1) When is it established?
(2) The hypothesis holds, and when it is deduced from it, the hypothesis also holds;
Then according to ① ②, it can be concluded that this proposition holds for all positive integers.
2. The second mathematical induction (string-valued induction)
Let be a proposition related to positive integers, if
(1) When is it established?
(2) The hypothesis holds, and when it is deduced from it, the hypothesis also holds;
Then according to ① ②, it can be concluded that this proposition holds for all positive integers.
3. Jump mathematical induction
Let be a proposition related to positive integers, if
(1) When is it established?
(2) The hypothesis holds, and when it is deduced from it, the hypothesis also holds;
Then according to ① ②, it can be concluded that this proposition holds for all positive integers.
4. Inverse mathematical induction
Let be a proposition related to positive integers, if
① It holds for infinite positive integers;
(2) From the establishment of the proposition, it can be concluded that the proposition is also established;
Then according to ① ②, it can be concluded that this proposition holds for all positive integers.
If the proposition is difficult to prove the establishment of infinite natural numbers, we can also consider two other forms of reverse mathematical induction:
Suppose I is a proposition related to a positive integer, if
(1) The proposition is correct;
(2) If it is not established, it will not be established;
Then according to ① ②, it can be concluded that this proposition holds for all positive integers.
Let Ⅱ be a proposition related to positive integers, if
(1), the proposition holds;
(2) If it is not established, it will not be established;
Then according to ① ②, it can be concluded that this proposition holds for all positive integers.
All the above discussions are complete induction. Incomplete induction is an important method to summarize general conclusions from a special perspective through experiments, observations, analysis, synthesis and abstraction. By using incomplete induction, through the calculation, observation and analysis of the first few terms of the series, we can infer its general term formula or the related properties of this series. When applying incomplete mathematical induction, the correctness of the conclusion must be proved by complete mathematical induction.
Using the skills of mathematical induction
1. Move the starting point
Some propositions are valid for all positive integers greater than or equal to 1, but the propositions themselves are also valid, and verification is easier than verification, so verification is used instead of verification; Similarly, the starting point can also be moved back appropriately, as long as the moved starting point is established and easy to verify.
2. The starting point has been raised
Some propositions need to use the properties of other special points when striding from one direction to another. At this time, it is often necessary to supplement and verify some special circumstances, so it is necessary to raise the starting point appropriately; Increasing the starting point can also better observe each unified form, thus proving it by mathematical induction.
3. Choose a suitable hypothesis.
The inductive hypothesis should not be limited to "the proposition holds when the hypothesis holds", but should adopt the first, second, jumping and reverse mathematical induction according to the meaning of the problem.
★ knowledge combing ★
1. Prove a proposition by mathematical induction in two steps. The first step is inductive basis (or recursive basis), and the second step is inductive recursion (or inductive hypothesis). These two steps are indispensable.
2. Many mathematical propositions related to natural numbers can be proved by mathematical induction, including identities, inequalities, formulas of general terms of series, divisibility, geometry and so on.
★ Break through difficulties ★
Emphasis: Understand the function of the two steps and prove some simple mathematical propositions by mathematical induction.
Difficulties: For different types of mathematical propositions, complete the recursion from k to k+ 1
Emphasis and difficulty: understand the principle of mathematical induction and use it correctly.
1. Proof without inductive hypothesis is not mathematical induction.
2. The starting point of induction is not necessarily 1.
3. "Induce-guess-prove" is an important way of thinking.