First, the basic method: look at the increase.

(1) If the increments are equal (this series 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.

Example: 4, 10, 16, 22, 28, find the nth place.

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). For example, the series 2, 5, 10, 17, 26, etc. Their growth rates are 3, 5, 7 and 9 respectively, indicating that the growth rate has the same increase. 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-th.

2. Find the total increment from bit 1 to bit n..

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

For example: 2, 5, 10, 17, 26, find the nth place.

Analysis: The rising rates of the series are 3, 5, 7 and 9 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. The total increment from bit 1 to bit n in the sequence is: the increment from bit 1 to bit 2 plus the increment from bit n- 1 to bit n, multiplied by the number of items increased from bit 1 to bit n in the sequence, and then divided by 2, that is:

[3+(2n-1) ]× (n-1) ÷ 2 = (n+1)× (n-1) = N2-1,so the nth digit is: 2.

Although this solution is annoying, it is a common way to solve this kind of problem. Of course, this problem can also be solved by the basic skills introduced below, and the method is much simpler.

(3) The rate of increase is not equal, but it is multiplied. There are general solutions, but this series should not be called junior high school students.

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, etc. 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.

Example 1: 1, n of 9,25,49 is (2n- 1)2.

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

Example 3: 2, 4, 8, 16, with increments of 2, 4, 8. The answer is related to the power of 2, that is, 2n.

(3) Some people can subtract the first number from each number at the same time to become a new series starting from the second number, and then find out the relationship between each number and its position by skill. Add the first number and go back to the original.

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

0,3,8, 1 5,24, 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.

(4) 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.

Example: 4 16, 36, 64, 144, 196, (the hundredth number).

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

(5) Like the techniques (3) and (4), some people can add, subtract, multiply or Divison the same number for each number (generally 1, 2, 3).

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