Explain the method of difference with examples?

Question: Explain the method of difference with examples?

 The method of difference is a technique used in mathematics to find patterns and relationships between numbers or variables. It involves subtracting the values of one term from the values of another term, typically in a sequence or series, to identify a pattern or relationship between them.

For example, consider the sequence of numbers: 1, 3, 6, 10, 15. By subtracting the values of consecutive terms, we get the sequence of differences: 2, 3, 4, 5. We can see that this sequence of differences is itself an arithmetic sequence with a common difference of 1. This tells us that the original sequence is a second-order sequence, meaning that the formula for the nth term involves a quadratic function.

Another example is finding the relationship between the volumes of cubes with different side lengths. If we take the difference between the volumes of cubes with side lengths 1, 2, 3, and 4, we get the sequence of differences: 7, 19, 37. We can see that this sequence is itself an arithmetic sequence with a common difference of 12. This tells us that the volume of a cube is a third-order sequence, meaning that the formula for the volume involves a cubic function.

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