等幂和差,又称幂和差,指同是n次幂的a与b的和与差,即 a n ± b n {\displaystyle a^{n}\pm b^{n}} 。 等幂和 当n为奇数时有: a n + b n = ( a + b ) ∑ r = 1 n a n − r ( − b ) r − 1 = ( a + b ) ( a n − 1 − a n − 2 b + . . . + b n − 1 ) {\displaystyle a^{n}+b^{n}=(a+b)\sum _{r=1}^{n}a^{n-r}(-b)^{r-1}=(a+b)(a^{n-1}-a^{n-2}b+...+b^{n-1})} 因尾项必须为+bn-1,故n不能取偶数。 等幂差 首项为1、公比为q的等比数列求和后得出: ∑ r = 1 n q r − 1 = 1 + q + . . . + q n − 1 = q n − 1 q − 1 {\displaystyle \sum _{r=1}^{n}q^{r-1}=1+q+...+q^{n-1}={\frac {q^{n}-1}{q-1}}} 设q=a/b,换算后即有: a n − b n = ( a − b ) ∑ r = 1 n a n − r b r − 1 = ( a − b ) ( a n − 1 + a n − 2 b + . . . + b n − 1 ) {\displaystyle a^{n}-b^{n}=(a-b)\sum _{r=1}^{n}a^{n-r}b^{r-1}=(a-b)(a^{n-1}+a^{n-2}b+...+b^{n-1})} 逆定理 ∑ r = 1 n a n − r b r − 1 = a n − 1 + a n − 2 b + . . . + b n − 1 = a n − b n a − b {\displaystyle \sum _{r=1}^{n}a^{n-r}b^{r-1}=a^{n-1}+a^{n-2}b+...+b^{n-1}={\frac {a^{n}-b^{n}}{a-b}}} ∑ r = 1 n a n − r ( − b ) r − 1 = a n − 1 − a n − 2 b + . . . + b n − 1 = a n + b n a + b {\displaystyle \sum _{r=1}^{n}a^{n-r}(-b)^{r-1}=a^{n-1}-a^{n-2}b+...+b^{n-1}={\frac {a^{n}+b^{n}}{a+b}}} 注意 ∑ r = 1 2 n − 1 a 4 n − 2 − 2 r b 2 r − 2 {\displaystyle \sum _{r=1}^{2n-1}a^{4n-2-2r}b^{2r-2}} 可进行因式分解,例如: a 4 + a 2 b 2 + b 4 = a 6 − b 6 a 2 − b 2 = ( a 3 + b 3 ) ( a 3 − b 3 ) ( a + b ) ( a − b ) = ( a 2 + a b + b 2 ) ( a 2 − a b + b 2 ) {\displaystyle a^{4}+a^{2}b^{2}+b^{4}={\frac {a^{6}-b^{6}}{a^{2}-b^{2}}}={\frac {(a^{3}+b^{3})(a^{3}-b^{3})}{(a+b)(a-b)}}=(a^{2}+ab+b^{2})(a^{2}-ab+b^{2})} 另解: a 4 + a 2 b 2 + b 4 = a 4 + 2 a 2 b 2 + b 4 − a 2 b 2 = ( a 2 + b 2 ) 2 − a 2 b 2 = ( a 2 + a b + b 2 ) ( a 2 − a b + b 2 ) {\displaystyle a^{4}+a^{2}b^{2}+b^{4}=a^{4}+2a^{2}b^{2}+b^{4}-a^{2}b^{2}=(a^{2}+b^{2})^{2}-a^{2}b^{2}=(a^{2}+ab+b^{2})(a^{2}-ab+b^{2})} 卢卡斯数列 等幂求和 Wikiwand in your browser!Seamless Wikipedia browsing. On steroids.Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.Wikiwand for ChromeWikiwand for Firefox
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.