1. Observe the relationship between the sum of cubes and the sum of squares: We know that any number of cubes can be divided into the form of the product of two numbers and the sum of another number, that is, a? =a？ × A. We can split the cube and formula into two square sum formulas, that is, n? =(n- 1)？ +n？ +(n+ 1)？ .
2. Prove by mathematical induction: We can prove the cubic sum formula by mathematical induction. Suppose n is an arbitrary positive integer, when n= 1, 1? = 1, the conclusion is valid; Suppose when n=k, k? =(k- 1)？ +k？ +(k+ 1)？ True, then when n=k+ 1, (k+ 1)? =k？ +2k+ 1=(k- 1)？ +k？ +(k+ 1)？ +4k, the conclusion is also valid. We can conclude that for any positive integer n, n? =(n- 1)？ +n？ +(n+ 1)？ Both are true.
3. Solve by using the cube sum formula: By using the cube sum formula, some problems of cube sum can be solved. For example, looking for 1? +2? +3? +…+ 10? The value of can be solved by adding the cubes of each number. Specifically, 1? = 1,2? =8,3? =27,4? =64,5? = 125,6? =2 16,7? =343,8? =5 12,9? =729, 10? = 1000。 1? +2? +3? +…+ 10? = 1+8+27+64+ 125+2 16+343+5 12+729+ 1000=2880。
Matters needing attention when deducing the formula:
1, Accuracy: When deriving a formula, ensure that every step of the derivation is accurate. This includes using correct symbols, definitions and theorems and ensuring that all steps are reversible. If the steps of derivation are wrong, then the final formula may also be wrong. Therefore, every step should be carefully checked when deducing the formula, and the derivation process should be carefully checked.
2. Completeness: When deriving the formula, you need to ensure that all steps are covered. This includes all assumptions, inferences and conclusions. If a step is omitted or omitted, it may affect the correctness of the final formula. Therefore, when deducing the formula, we should list all the steps in detail and ensure that the logical relationship between them is complete.
3. Logic: Formulas should follow the principle of logical reasoning. This includes reasoning with known conditions, assumptions and theorems to ensure that every step of reasoning is reasonable. If a reasoning step is not logical, then the final formula may be incorrect. Therefore, when deducing the formula, we should follow the logical principle to ensure that every step of reasoning is logical.