How to verify arithmetic progression by mathematical induction

Mathematical induction is a special method to prove the proposition of natural number n in mathematics. It is mainly used to study mathematical problems related to positive integers, and is often used to prove the establishment of equality and the general term formula of sequence in high school mathematics.

Mathematical induction includes the following types:

(A) the first mathematical induction

Generally speaking, to prove that a proposition P(n) is related to the natural number n, there are the following steps:

(1) proves that the proposition holds when n takes the first value n0. The value of n0 is generally 0 or 1, but there are some special cases.

(2) Assuming that the proposition holds when n=k(k≥n0, k is a natural number), it is proved that the proposition also holds when n=k+ 1.

Synthesizing (1)(2), the proposition P(n) holds for all natural numbers n(≥n0).

(B) the second mathematical induction

For a proposition P(n) related to natural numbers,

(1) Verify that P(n) holds when n=n0 and n=n 1;

(2) Assuming that the proposition holds when n≤k, on this basis, the proposition of n=k+ 1 also holds.

Synthesizing (1)(2), the proposition P(n) holds for all natural numbers n(≥n0).

(3) Reverse induction

Also known as backward induction

(1) Verify that the proposition P(n) holds for an infinite number of natural numbers (the infinite number of natural numbers can be numbers in an infinite sequence, for example, the proof of arithmetic geometric inequality is 2 k, and k ≥1);

(2) Assume that P(k+ 1)(k≥n0) holds, and then deduce that P(k) holds.

Synthesizing (1)(2), the proposition P(n) holds for all natural numbers n(≥n0);

Spiral induction

For two propositions P(n) and Q(n) related to natural numbers,

(1) Verify that P(n) holds when n=n0;

(2) suppose p (k) (k >; N0) holds, Q(k) holds, assuming Q(k) holds, P(k+ 1) holds;

Composition (1)(2) is applicable to all natural numbers n(≥n0), P(n) and Q(n).

Here is an example for you to understand:

Q: Is there a arithmetic progression?

{an}, so for any natural number n, the equation:? a 1+2 a2+3 a3+…+nan = n(n+ 1)(n+2)？ Both are true, prove your conclusion.

Analysis: adopt the thinking method from special to general,

Find {an} when n= 1, 2, 3, and then prove generality.