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我们知道,当n是正整数时有即x~n-y~n能被x-y整除; 当n是正奇数时有 即x~n+y~n能被x+y整除. 我们感兴趣的是二项公式具有可整除性的特点,它能巧妙应用于证明等比数列前n项和的公式,数列递推通项公式,解某一类特殊方程,多项式因式分解,证某一类不等式等。 例1 证明等比数列前n项之和的公式 应用二项公式可以给出一种简捷的证法。 证明:设等比数列为 则 上式两边乘以(1-q), 得(1-q)S_n=a_1(1-q~n), ∴S_n=a_1(1-q~n)/1-q (q≠1).
We know that when n is a positive integer, that is, x~ny~n can be divided by xy; when n is a positive odd number, that is, x~n+y~n can be divisible by x+y. We are interested in the binomial formula with The characteristics of divisibility can be applied to the formulae that prove the n-sum of the equal-ratio series, the formula of the series recursive term, the solution of a certain kind of special equation, the polynomial factorization, and the proof of certain inequalities. Example 1 Formula to prove the sum of the first n terms of the geometric sequence The application of the binomial formula gives a simple proof. Prove that if we set the equal-ratio sequence to be (1-q), then (1-q)S_n=a_1(1-q~n), ∴S_n=a_1(1-q~n)/1- q (q≠1).