对正整数 n＞5 有 n^5=aC(n,5)+bC(n,4)+cC(n,3)+dC(n,2)+eC(n,1)，求 a+b+c+d+e

wintex

对正整数 n＞5 有 n^5=aC(n,5)+bC(n,4)+cC(n,3)+dC(n,2)+eC(n,1)，求 a+b+c+d+e

Treenewbee

待定系数法：

\[n^5=aC_n^5+bC_n^4+cC_n^3+dC_n^2+eC_n^1\]
\[=\frac{a n^5}{120}-\frac{a n^4}{12}+\frac{b n^4}{24}+\frac{7 a n^3}{24}-\frac{b n^3}{4}+\frac{c n^3}{6}-\frac{5 a n^2}{12}+\frac{11 b n^2}{24}-\frac{c n^2}{2}+\frac{d n^2}{2}+\frac{a n}{5}-\frac{b n}{4}+\frac{c n}{3}-\frac{d n}{2}+e n\]

fungarwai

\(n^5=a\binom{n}{5}+b\binom{n}{4}+c\binom{n}{3}+d\binom{n}{2}+e\binom{n}{1}+f\binom{n}{0}\)
\(0^5=f\)
\(1^5=e+f\)
\(2^5=d+2e+f\)
\(3^5=c+3d+3e+f\)
\(4^5=b+4c+6d+4e+f\)
\(5^5=a+5b+10c+10d+5e+f\)

\(1^5-0^5=e\)
\(2^5-1^5=d+e\)
\(3^5-2^5=c+2d+e\)
\(4^5-3^5=b+3c+3d+e\)
\(5^5-4^5=a+4b+6c+4d+e\)

\(2^5-2\times 1^5+0^5=d\)
\(3^5-2\times 2^5+1^5=c+d\)
\(4^5-2\times 3^5+2^5=b+2c+d\)
\(5^5-2\times 4^5+3^5=a+3b+3c+d\)

\(3^5-3\times 2^5+3\times 1^5-0^5=c\)
\(4^5-3\times 3^5+3\times 2^5-1^5=b+c\)
\(5^5-3\times 4^5+3\times 3^5-2^5=a+2b+c\)

\(4^5-4\times 3^5+6\times 2^5-4\times 1^5+0^5=b\)
\(5^5-4\times 4^5+6\times 3^5-4\times 2^5+1^5=a+b\)

\(5^5-5\times 4^5+10\times 3^5-10\times 2^5+5\times 1^5-0^5=a\)

luyuanhong

luyuanhong