1, increase or decrease observation
Observe two adjacent numbers in the sequence, and judge whether the difference between them is increasing or decreasing, so as to determine the law of the sequence. For example, 1, 3, 5, 7, 9 ... is a arithmetic progression with a difference of 2.
2. Calculate the sum or difference of adjacent numbers.
For arithmetic progression or geometric progression, the law can be found by calculating the sum or difference of two adjacent numbers. For example, the sum of two adjacent numbers of 1, 3,5,7 ... is 4,6,8, ... and the difference between two adjacent numbers 2,4,6,8 ... is 2,2,2.
3, identify the periodic law
Some series will appear in a certain period, so we should pay attention to identify the laws of these periods. For example, 1, 2, 3, 4, 5, 6, 7, 8 ... is a cyclic sequence with a period of 7.
4. Decomposition sequence
For some complex series, we can try to decompose them into simple series or combination rules. For example:1+2+3 =1× 2× 3; 1+2+3+4=2×3×4; 1+2+3+4+5=3×5×6……。
5, looking for symmetry
Some series are symmetrical, such as 1, 3, 5, 7 … symmetric sequence, and the first number is equal to the last number. Similarly, there are some reverse sequences and so on.
The general law of discovering laws in mathematics can be summarized as follows:
1, observation rule: to observe the rule of a series or graph, we can start with the size, shape and color of the series or graph.
2. Summary: Through observation, analysis and reasoning, we can summarize the laws of series or figures.
3. Verify the law: for the found law, you can substitute it into a known series or graph for verification to see if it conforms to the law.
4. Use the law: by looking for the law, solve some mathematical problems related to series or figures, and improve your mathematical literacy.
Finding the law is not an easy task, which requires patience, care and serious thinking. At the same time, the discovery of laws also requires certain methods and skills, which can be discovered through observation, analysis and reasoning.