The definition is that in a certain kind of problem, only a few limited situations are tested and general conclusions are drawn. This inductive reasoning method is called "incomplete induction".

Characteristics (1) local rationality;

(b) overall inaccuracy.

Man always touches the world from some individual aspects, and it is impossible to research his whole and whole.

(1) This inductive reasoning can have certain reference value and even guiding significance for the local real world for the time being.

(2) This inductive reasoning promotes the development of science. As we all know, a reasonable conclusion is not necessarily a correct conclusion; However, the lack of rigorous reasoning does not mean that the conclusion must be wrong, which urges everyone to strictly demonstrate or find counterexamples to overthrow it.

It is no exaggeration to say that the development of science is bound to go through the step of incomplete induction.

Some basic facts-reasonable induction, bold assumption-careful argumentation, part and concrete-(deduction and demonstration)-whole and abstract-(guidance and application)-part and concrete.

For example: ① We change from 1 = 1 2, 1+3 = 2 2, 1+3+5 = 3 2, 1+3+7 = 42,1.

Through analysis and research, the conclusion of 1+3+5+ ...+(2n-1) = n 2.

② For quadratic function f (x) = x 2+x 41,

There are F (0) = 4 1, F (1) = 43, F (2) = 47, F (3) = 53, F (4) = 6 1, and f (6) = 7/kloc-0.

We conclude that f(n) is a prime number.

The charm of incomplete induction can be seen from Goldbach's conjecture. Even if Goldbach's conjecture is finally denied, the function of incomplete induction cannot be denied.

complete induction

Adaptive question type is a mathematical proof method that adapts to arithmetic proposition (any positive integer starting with n≥N).

There are three steps to prove an arithmetic proposition by mathematical induction.

▲ ① Preliminary verification

Using mathematical induction to prove that "any positive integer starting from positive integer n holds", the first step is to start from the proposition that "it must hold" when n=N here;

▲ ② general assumption

Assuming that n=k(≥N), the proposition also holds;

▲ ③ Progressive recursion

Using the "preliminary verification" of ① and the "general assumption" of ②, it is strictly deduced that the proposition also holds when n=k+ 1

After the logical meaning verifies that the initial proposition of n=N is "necessarily true", there is no need to exhaust all kinds of situations, such as n = n+ 1, n = n+2, ... because of the existence of ② and ③, there is no doubt:

If n=N is correct, then n=N+ 1 must be correct;

There must be the correctness when n=N+ 1 and when n=N+2;

There must be the correctness when n=N+2 and when n=N+3; ... even when n is any positive integer greater than n 。