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杨辉恒等式即现行高中数学教材中所述组合数的第二个基本性质:C_(n-1)~(i-1)+C_(n-1)~i=C_n~i(1≤i≤n-1)(1) 我们可以结合等差数列将其推广为定理设a_0,a_1,…,a_n是一个等差数列,则当0≤i≤n时,恒有 a_iC_n~i=a_nC_(n-1)~(i-1)+a_0C_(n-1)~i(2) 证明:当i=0或n时,按规定有C_(n-1)~n=0,C_(n-1)~(-1)=0,此时,(2)式显然成立。当1≤i≤n-1时,设等差数列a_0,a_1,…,a_n的公差为d,则a_i=a_0+id (0≤i≤n),于是
The Yang Huiheng equation is the second basic property of the combination number mentioned in the current high school mathematics textbook: C_(n-1)~(i-1)+C_(n-1)~i=C_n~i(1≤i≤n -1)(1) We can combine it with the arithmetic sequence to generalize it to the theorem Let a_0,a_1,...,a_n be an arithmetic progression, then when 0≤i≤n, we have a_iC_n~i=a_nC_(n- 1)~(i-1)+a_0C_(n-1)~i(2) Prove that when i=0 or n, there are C_(n-1)~n=0, C_(n-1) as required ~ (-1) = 0, this time, (2) is clearly established. When 1 ≤ i ≤ n-1, it is assumed that the tolerance of the arithmetic progression a_0, a_1,..., A_n is d, then a_i=a_0+id (0≤i≤n), and thus