(1) If the increase is equal (this is actually arithmetic progression): compare each number with its previous number. If the increments are equal, the nth number can be expressed as: a+(n- 1)b, where a is the first digit of the sequence, b is the increment, and (n- 1)b is the total increment from the first digit to the nth digit. Then simplify the algebraic expression a+(n-1) b.

For example: 4, 10, 16, 22, 28 ... Find the nth number.

Analysis: Starting from the second bit, each bit is 6 more than the previous bit, and the added phase is 6, so the nth bit is: 4+(n- 1) × 6 = 6n-2.

(2) If the increase rate is not equal, but the increase rate is the same (that is, the increase rate is equal, that is, the increase rate is arithmetic progression). If the growth rates are 3, 5, 7 and 9 respectively, it means that the growth rate has increased by the same amount. The number of the nth bit of this series also has a general solution.

The basic idea is: 1, find the increment from n- 1 to n;

2. Find the total increase from 1 to the nth place;

3. The 1 bit of the sequence plus the total increment is the nth bit.

For example: 2,5, 10, 17 ..., find the nth digit.

Analysis: The rising rates of the series are 3, 5 and 7 respectively, and the rising rates increase by the same amount. Then, the increment from the number n- 1 to the number n is: 3+2×(n-2)=2n- 1, and the total increment is:

の3+(2n- 1)の×(n- 1)÷2 =(n+ 1)×(n- 1)= N2- 1

Therefore, the nth digit is: 2+ n2- 1= n2+ 1.

Although this solution is annoying, the general solution to this kind of problem, of course, can also be solved by other techniques or through analysis and observation, and the method is much simpler.

(3) The growth rate is not equal, but it increases year-on-year, that is, the growth rate increases geometrically, such as 2, 3, 5, 9, 17, and the growth rate is 1, 2, 4, 8.

(3) Unequal increase, unequal increase (that is, unequal increase). There is probably no general solution to this kind of problem, only the method of analysis and observation. But this kind of problem includes the second kind of problem. If we use the method of analysis and observation, there are some skills.

Second, basic skills

(1) serial number: the problem of finding regularity is usually to give a series of quantities in a certain order, which requires us to find general laws 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.

For example, observe the following numbers: 0, 3, 8, 15, 24, ... try to write the number 100th according to this rule.

To solve this problem, we can first find the general law, and then use this law to calculate the number 100. Let's compare the related quantities together:

Numbers given: 0, 3, 8, 15, 24, ...

Serial number: 1, 2, 3, 4, 5,.

It is easy to find that each term of the known number is equal to the square of its serial number minus 1. So the nth term is n2- 1, and the first term 100 is 1002- 1.

(2) Common factor method: multiply each number by the least common factor, and then find the law to see if it is related to n2, n3, 2n, 3n, or 2n, 3n.

For example: 1, 9, 25, 49, (), (), and the nth one is (2n- 1)2 (3). See the example:

A: 2, 9, 28, 65 ... the increase is 7, 19, 37 ... the increase is 12, 18. The answer is related to 3 and ............ is: n3+ 1.

B: 2, 4, 8, 16 ... The increase is 2, 4, 8 ... The answer is related to the power of 2, which is 2n.

(4) Some people can subtract the first number from each number at the same time to form a new series starting from the second number, and then use the skills of (1), (2) and (3) to find out the relationship between each number and its position. Then add the first number to the found law and restore it to its original appearance.

Example: 2, 5, 10, 17, 26 ... and subtract 2 to get a new series:

0、3、8、 15、24……,

Serial number: 1, 2, 3, 4, 5

According to the analysis and observation, the nth term of the new series is n2- 1, so the nth term of the series in the problem is: (N2- 1)+2 = N2+ 1.

(5) Some can add, multiply or divide each number at the same time to form a new series, and then find out the law again and return to the original point.

For example: 4, 16, 36, 64, 144, 196, …? (hundredth digit)

Divided by 4, we can get a new series: 1, 4, 9, 16…, which is obviously the square of digits.

(6) Like techniques (4) and (5), some people can add, subtract, multiply or Divison the same number for each number (generally 1, 2,3). Of course, it is more likely to do addition or subtraction at the same time, and it is less common to do multiplication or division at the same time.

(7) Observe whether the odd and even digits of a series can be divided into two series, and then look for the rules respectively.

Third, the basic steps

1, first see whether the increase is equal, if equal, use the basic method (1) to solve the problem.

2. If they are not equal, comprehensively apply skills (1), (2) and (3) to find the law.

3. If not, use skills (4), (5) and (6) to transform into a new series, and then use skills (1), (2) and (3) to find out the law of the new series.

4. Finally, if the increase rate increases by the same amount, use the basic method (2) to solve the problem.

Summarize a lot by yourself, and I'll test your method.