It is proved that the proposition holds when n= 1.
Assuming that the proposition holds when n=m, it can be deduced that the proposition also holds when n=m+ 1 (m stands for any natural number)
The principle of this method lies in: first, prove that the proposition is valid at a certain initial value, and then prove that the process from one value to the next is valid. When these two points are proved, then any value can be deduced by repeatedly using this method.
Mathematical induction has strict requirements on the form of solving problems. In the process of solving problems by mathematical induction,
Step 1: Verify that n holds when it takes the first natural number.
Step 2: Assume that n=k holds, and then deduce it according to the conditions of verification and hypothesis. In the following derivation process, n=k+ 1 cannot be directly substituted into the assumed original formula.
The last step is to summarize the statement.
It should be emphasized that the two steps of mathematical induction are both very important and indispensable, otherwise the following absurd proof may be obtained:
Proof 1: All horses are one color.
First of all, in the first step, this proposition holds for n= 1, that is, when there is only 1 horse, there is only one color of the horse.
In the second step, assume that this proposition holds true for n, that is, assume that any n horses are one color. So when we have n+ 1 horses, we might as well number them:
1，2，3……n，n+ 1
For (1, 2...n) these horses, we can get from our hypothesis that they are all the same color;
For (2,3 ... n, n+ 1) these horses, we can also get that they are one color;
Because there are (2, 3, ... n) horses in two groups, it can be concluded that the n+ 1 horses are all of the same color.
The error of this proof comes from the second inference: when n= 1 and n+ 1=2, at this time, the number of horses is only 1 and 2, so the two groups are (1) and (2)-they do not intersect, so the second inference is wrong. The second step of mathematical induction requires that the process of n→n+ 1 holds for all numbers of n= 1, 2, 3 ... and the above proof is just like the gap between the first block and the second block of a domino is too big, which overturns the first block, but not the second block. Even though we know that the fall of the second block will push down the third block and so on, this process has been interrupted between the first block and the second block.