Solve it by mathematical induction.

For the first time, a rope was folded in half and cut from the middle; The first power of 2+1 = 3;

The second time, a rope was folded in half into four sections and cut from the middle; Quadratic of 2+1= 5;

The third time, a rope was folded in half into 8 sections and cut from the middle; 2 to the third power+1= 9;

Fold it 8 times, and the answer is 2 to the 8th power+1=257.

So the formula is: fold n times, that is, the root of (2 to the nth power+1).

Extended data:

The simplest and most common mathematical induction is to prove that a proposition holds when n is equal to any natural number. Proof is divided into the following two steps:

1. Prove that the proposition holds when n= 1.

2. 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)

Find a routine method:

1, serial number: to find a regular topic, a series of quantities arranged in a certain order are usually given, and we are required to find a general rule according to these known quantities. Find out the rule, usually the serial number of the package. Therefore, it is easier to find the mystery by comparing variables with serial numbers.

2. Fibonacci sequence method: each number is the sum of the first two numbers.

3. Arithmetic series method: the difference between every two numbers is equal.

4. Tabbing method: You can look at it every once in a while to see what the relationship between the separated numbers is, for example, 14, 1, 12, 3, 10, 5, odd items become arithmetic progression, even items also become arithmetic progression, and then you have to fill in 8.

5. Incremental method: See if the difference between every two numbers is arithmetic progression, such as 1, 4, 8, 13, 19, and the difference between every two numbers is 3, 4, 5 and 6 respectively, so the next difference should be 7, that is, 26.