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受欢迎的代数 >

展开 (1+x/2)^{18}

解答

展开

(1 + \frac{x}{2})^{18}

解答

1 + 9 x + \frac{153 x ^{2}}{4} + 102 x^{3} + \frac{765 x ^{4}}{4} + \frac{1071 x ^{5}}{4} + \frac{4641 x ^{6}}{16} + \frac{1989 x ^{7}}{8} + \frac{21879 x ^{8}}{128} + \frac{12155 x ^{9}}{128} + \frac{21879 x ^{10}}{512} + \frac{1989 x ^{11}}{128} + \frac{4641 x ^{12}}{1024} + \frac{1071 x ^{13}}{1024} + \frac{765 x ^{14}}{4096} + \frac{51 x ^{15}}{2048} + \frac{153 x ^{16}}{65536} + \frac{9 x ^{17}}{65536} + \frac{x ^{18}}{262144}

求解步骤

(1 + \frac{x}{2})^{18}

使用二项式定理:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 1, b = \frac{x}{2}

= i = 0 \sum 18 (i 18) \cdot 1^{(18 - i)} (\frac{x}{2})^{i}

展开求和

= \frac{18 !}{0 ! ( 18 - 0 ) !} \cdot 1^{18} (\frac{x}{2})^{0} + \frac{18 !}{1 ! ( 18 - 1 ) !} \cdot 1^{17} (\frac{x}{2})^{1} + \frac{18 !}{2 ! ( 18 - 2 ) !} \cdot 1^{16} (\frac{x}{2})^{2} + \frac{18 !}{3 ! ( 18 - 3 ) !} \cdot 1^{15} (\frac{x}{2})^{3} + \frac{18 !}{4 ! ( 18 - 4 ) !} \cdot 1^{14} (\frac{x}{2})^{4} + \frac{18 !}{5 ! ( 18 - 5 ) !} \cdot 1^{13} (\frac{x}{2})^{5} + \frac{18 !}{6 ! ( 18 - 6 ) !} \cdot 1^{12} (\frac{x}{2})^{6} + \frac{18 !}{7 ! ( 18 - 7 ) !} \cdot 1^{11} (\frac{x}{2})^{7} + \frac{18 !}{8 ! ( 18 - 8 ) !} \cdot 1^{10} (\frac{x}{2})^{8} + \frac{18 !}{9 ! ( 18 - 9 ) !} \cdot 1^{9} (\frac{x}{2})^{9} + \frac{18 !}{10 ! ( 18 - 10 ) !} \cdot 1^{8} (\frac{x}{2})^{10} + \frac{18 !}{11 ! ( 18 - 11 ) !} \cdot 1^{7} (\frac{x}{2})^{11} + \frac{18 !}{12 ! ( 18 - 12 ) !} \cdot 1^{6} (\frac{x}{2})^{12} + \frac{18 !}{13 ! ( 18 - 13 ) !} \cdot 1^{5} (\frac{x}{2})^{13} + \frac{18 !}{14 ! ( 18 - 14 ) !} \cdot 1^{4} (\frac{x}{2})^{14} + \frac{18 !}{15 ! ( 18 - 15 ) !} \cdot 1^{3} (\frac{x}{2})^{15} + \frac{18 !}{16 ! ( 18 - 16 ) !} \cdot 1^{2} (\frac{x}{2})^{16} + \frac{18 !}{17 ! ( 18 - 17 ) !} \cdot 1^{1} (\frac{x}{2})^{17} + \frac{18 !}{18 ! ( 18 - 18 ) !} \cdot 1^{0} (\frac{x}{2})^{18}

化简

\frac{18 !}{0 ! ( 18 - 0 ) !} \cdot 1^{18} (\frac{x}{2})^{0} : 1

化简

\frac{18 !}{1 ! ( 18 - 1 ) !} \cdot 1^{17} (\frac{x}{2})^{1} : 9 x

化简

\frac{18 !}{2 ! ( 18 - 2 ) !} \cdot 1^{16} (\frac{x}{2})^{2} : \frac{153 x ^{2}}{4}

化简

\frac{18 !}{3 ! ( 18 - 3 ) !} \cdot 1^{15} (\frac{x}{2})^{3} : 102 x^{3}

化简

\frac{18 !}{4 ! ( 18 - 4 ) !} \cdot 1^{14} (\frac{x}{2})^{4} : \frac{765 x ^{4}}{4}

化简

\frac{18 !}{5 ! ( 18 - 5 ) !} \cdot 1^{13} (\frac{x}{2})^{5} : \frac{1071 x ^{5}}{4}

化简

\frac{18 !}{6 ! ( 18 - 6 ) !} \cdot 1^{12} (\frac{x}{2})^{6} : \frac{4641 x ^{6}}{16}

化简

\frac{18 !}{7 ! ( 18 - 7 ) !} \cdot 1^{11} (\frac{x}{2})^{7} : \frac{1989 x ^{7}}{8}

化简

\frac{18 !}{8 ! ( 18 - 8 ) !} \cdot 1^{10} (\frac{x}{2})^{8} : \frac{21879 x ^{8}}{128}

化简

\frac{18 !}{9 ! ( 18 - 9 ) !} \cdot 1^{9} (\frac{x}{2})^{9} : \frac{12155 x ^{9}}{128}

化简

\frac{18 !}{10 ! ( 18 - 10 ) !} \cdot 1^{8} (\frac{x}{2})^{10} : \frac{21879 x ^{10}}{512}

化简

\frac{18 !}{11 ! ( 18 - 11 ) !} \cdot 1^{7} (\frac{x}{2})^{11} : \frac{1989 x ^{11}}{128}

化简

\frac{18 !}{12 ! ( 18 - 12 ) !} \cdot 1^{6} (\frac{x}{2})^{12} : \frac{4641 x ^{12}}{1024}

化简

\frac{18 !}{13 ! ( 18 - 13 ) !} \cdot 1^{5} (\frac{x}{2})^{13} : \frac{1071 x ^{13}}{1024}

化简

\frac{18 !}{14 ! ( 18 - 14 ) !} \cdot 1^{4} (\frac{x}{2})^{14} : \frac{765 x ^{14}}{4096}

化简

\frac{18 !}{15 ! ( 18 - 15 ) !} \cdot 1^{3} (\frac{x}{2})^{15} : \frac{51 x ^{15}}{2048}

化简

\frac{18 !}{16 ! ( 18 - 16 ) !} \cdot 1^{2} (\frac{x}{2})^{16} : \frac{153 x ^{16}}{65536}

化简

\frac{18 !}{17 ! ( 18 - 17 ) !} \cdot 1^{1} (\frac{x}{2})^{17} : \frac{9 x ^{17}}{65536}

化简

\frac{18 !}{18 ! ( 18 - 18 ) !} \cdot 1^{0} (\frac{x}{2})^{18} : \frac{x ^{18}}{262144}

= 1 + 9 x + \frac{153 x ^{2}}{4} + 102 x^{3} + \frac{765 x ^{4}}{4} + \frac{1071 x ^{5}}{4} + \frac{4641 x ^{6}}{16} + \frac{1989 x ^{7}}{8} + \frac{21879 x ^{8}}{128} + \frac{12155 x ^{9}}{128} + \frac{21879 x ^{10}}{512} + \frac{1989 x ^{11}}{128} + \frac{4641 x ^{12}}{1024} + \frac{1071 x ^{13}}{1024} + \frac{765 x ^{14}}{4096} + \frac{51 x ^{15}}{2048} + \frac{153 x ^{16}}{65536} + \frac{9 x ^{17}}{65536} + \frac{x ^{18}}{262144}

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