In junior high school mathematics examination, the problem of finding the law of sequence often appears. This paper discusses the methods to solve these problems: 1. Basic method-Look at the increments (1) If the increments are 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 a series. (n- 1)b is the total increment from the first number to the nth number. Then simplify the algebraic expression a+(n-1) b. For example, 4, 10, 16, 22, 28 ..., find the nth digit. Analysis: the second type. The nth bit is: 4+(n- 1) × 6 = 6n-2 (2). If the increments are not equal, the increments increase by the same magnitude (i.e. the increments are equal, i.e. the increments are arithmetic progression). If the increase amounts are 3, 5, 7 and 9 respectively, it means that the increase amounts are increased by the same amplitude. There is another number in the nth digit of this series. 2. Find the total increase from 1 to the nth place; 3. the number 1 plus the total increment of the series is the number n. For example, 2, 5, 10, 17 ..., find the number n. Analysis: the increments of the series are 3, 5 and 7 respectively, and the increments are the same. Then, the number n-65438 of the total increment of the sequence is: [3+(2n-1) ]× (n-1) ÷ 2 = (n+1)× (n-1) = N2- (3) The growth rate is not equal, but it increases year-on-year, that is, the growth rate is geometric series, for example, the growth rate of 2, 3, 5, 9, 17 is 1, 2, 4, 8. (3) Unequal growth rates, unequal growth rates (that is, unequal growth rates). If you use the analytical observation method, there are also some skills. Second, the basic skills (1) serial number: the topic of finding laws is usually to give a series of quantities in a certain order, which requires us to find general laws according to these known quantities. The laws found usually include serial numbers. Therefore, it is easier to find the mystery by comparing variables with serial numbers. For example, observe the following numbers: 0, 3, 8. ..... 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 100th. We compare the related quantities together: the numbers given are: 0, 3, 8, 15, 24, and ................. 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 bit by the least common factor, and then find the law to see if it is the same as n2 and n3. (), the nth one is (2n- 1)2 (3). See examples: A: 2, 9, 28, 65. The increase is 7, 19, 37. The increase is 12, 18. The answer is related to 3, namely n3+ 1b: 2, 4, 8, 16. The growth rate 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. Then use the skills of (1), (2) and (3) to find out the relationship between the number and position of each number. Then add the first number to the found rule and restore it to its original appearance. Example: 2, 5, 10, 17, 26 ... and subtract 2 to get a new series: 0, 3, 8, 6550. The nth term of the new series is n2- 1, so the nth term of the series in the question is: (N2- 1)+2 = N2+ 1 (5). Some numbers can be added at the same time, or multiplied or divided by the first number to form a new series. Then, find out the law and return to nature. , 144, 196,… ? Divide (the hundredth number) by 4 to get a new series: 1, 4, 9, 16…, which is obviously the square of digits. (6) Techniques like (4) and (5), some of which can add, subtract, multiply and divide each number by the same number (generally 65438), can you divide the odd and even digits of a series into two series, and then find the rules respectively? 3. Basic steps 1. First of all, look at whether the increase is equal. If so, solve the problem in a basic way. 2. If not, comprehensively use skills (1), (2) and (3) to find the law. If not, use technique (4) and then use techniques (1), (2) and (3) to find out the law of the new sequence. 4. Finally, if the increase rate increases at the same speed, use the basic method (2) to solve the problem. 4. Example of exercises 1: The rules for solving a junior high school math problem are 0, 3, 8, 15, 24, ... (2) What are the relations between the second group and the third group respectively? (3) Take the seventh number of each group and find the sum of these three numbers? 2. Observe the following two lines: 2, 4, 8, 16, 32, 64, ... (1) 5, 7,1,19, 35, 67 ... 4.32-1 2 = 8×152-32 = 8× 272-52 = 8× 3 ... Write an equation with the square difference of 888 for two continuous technologies with an algebraic expression with n. For the table1,