Common methods of mathematical induction

The first mathematical induction. Make sure that the expression is valid in all natural numbers, or that another form is valid in an infinite sequence. The generalized form of mathematical logic and computer science points out that the expressions that can be found are equivalent.

The second backward induction. It is proved that the sum formula and general formula of the first n terms of the sequence are valid.

The third spiral induction. Prove inequalities related to natural numbers.

The principle of mathematical induction is: first, it is proved that the proposition is valid at a certain starting value (positive integer or natural number), and then it is proved that the process of deriving from any value to the next value is effective. When these two points are proved, they can all be verified by using this method repeatedly. This method can be compared with dominoes.

For example, there is a long string of dominoes standing upright. If possible:

1. Prove that the first domino will fall.

2. Prove that as long as any domino falls, the next domino adjacent to it will also fall.

Then it can be concluded that all dominoes will fall.

In the college entrance examination, some topics related to mathematical induction are often combined with series. When solving problems related to sequence, teachers can guide students to make assumptions first and then prove them by mathematical induction, so as to clearly sort out the problem-solving ideas and get the correct answers.

For example, the sequence {an} is known, where a2 = 6, (an+1+an-1)/(an+1-an+1) = n.

(1) found a 1, a3, a4.

(2) Find the general term formula of the sequence.

For the first quiz, substitute n= 1, n=2 and n=3 into the above formula respectively.

(A2+a 1- 1)/(A2-a 1+ 1)= 1①；

(a3+a2- 1)/(a3-a2+ 1)= 2②；

(a4+a3- 1)/(a4-a3+ 1)= 3③；

Substituting the known condition a2=6 into ① ② formula, we can get a 1= 1, a3= 15, and then substitute the value of a3 into ③ formula to get a4=28, thus solving the problem of the first quiz. This kind of problem does not require students' thinking and logical ability. In the process of solving problems, it is not difficult for students to work out the answers.

As for the second quiz, the known condition is the specific values of the first four items in the series, and there is no other information except a recursive formula. At this point, teachers can guide students to summarize the laws of known information, guess the general terms of the series through the structural characteristics of the first four items, and then use mathematical induction to make assumptions and then prove them, and finally get the answer.

On this topic, we can write the first four terms of the series as a 1= 1* 1, a2=2*3, a3=3*5 and a4=4*7 respectively. Observing its structural characteristics, we can find that the values of the first four terms can be expressed as the product of a positive integer and an odd number. Namely: a1=1* (2 *1-1), a2=2*(2*2- 1), a3 = 3 * (2 * 3-/kloc-0).