展开 1000(1+i/(12))^{12}

解答

展开

1000 (1 + \frac{i}{12})^{12}

\frac{1625 + 3000 i}{3} + (\frac{6875}{288} - \frac{9625}{31104} + \frac{6875}{5971968} - \frac{1375}{1289945088} + \frac{125}{1114512556032}) + (- \frac{6875}{54} + \frac{1375}{432} - \frac{1375}{62208} + \frac{6875}{161243136} - \frac{125}{2 ^{17} \cdot 59049}) i

求解步骤

1000 (1 + \frac{i}{12})^{12}

展开

(1 + \frac{i}{12})^{12} : 1 + i + \frac{11 i ^{2}}{24} + \frac{55 i ^{3}}{432} + \frac{55 i ^{4}}{2304} + \frac{11 i ^{5}}{3456} + \frac{77 i ^{6}}{248832} + \frac{11 i ^{7}}{497664} + \frac{55 i ^{8}}{47775744} + \frac{55 i ^{9}}{1289945088} + \frac{11 i ^{10}}{10319560704} + \frac{i ^{11}}{1 2 ^{10}} + \frac{i ^{12}}{1 2 ^{12}}

= 1000 (1 + i + \frac{11 i ^{2}}{24} + \frac{55 i ^{3}}{432} + \frac{55 i ^{4}}{2304} + \frac{11 i ^{5}}{3456} + \frac{77 i ^{6}}{248832} + \frac{11 i ^{7}}{497664} + \frac{55 i ^{8}}{47775744} + \frac{55 i ^{9}}{1289945088} + \frac{11 i ^{10}}{10319560704} + \frac{i ^{11}}{1 2 ^{10}} + \frac{i ^{12}}{1 2 ^{12}})

打开括号

= 1000 \cdot 1 + 1000 i + 1000 \cdot \frac{11 i ^{2}}{24} + 1000 \cdot \frac{55 i ^{3}}{432} + 1000 \cdot \frac{55 i ^{4}}{2304} + 1000 \cdot \frac{11 i ^{5}}{3456} + 1000 \cdot \frac{77 i ^{6}}{248832} + 1000 \cdot \frac{11 i ^{7}}{497664} + 1000 \cdot \frac{55 i ^{8}}{47775744} + 1000 \cdot \frac{55 i ^{9}}{1289945088} + 1000 \cdot \frac{11 i ^{10}}{10319560704} + 1000 \cdot \frac{i ^{11}}{1 2 ^{10}} + 1000 \cdot \frac{i ^{12}}{1 2 ^{12}}

化简

1000 \cdot 1 + 1000 i + 1000 \cdot \frac{11 i ^{2}}{24} + 1000 \cdot \frac{55 i ^{3}}{432} + 1000 \cdot \frac{55 i ^{4}}{2304} + 1000 \cdot \frac{11 i ^{5}}{3456} + 1000 \cdot \frac{77 i ^{6}}{248832} + 1000 \cdot \frac{11 i ^{7}}{497664} + 1000 \cdot \frac{55 i ^{8}}{47775744} + 1000 \cdot \frac{55 i ^{9}}{1289945088} + 1000 \cdot \frac{11 i ^{10}}{10319560704} + 1000 \cdot \frac{i ^{11}}{1 2 ^{10}} + 1000 \cdot \frac{i ^{12}}{1 2 ^{12}} : \frac{3000 i + 1625}{3} - \frac{6875 i}{54} + \frac{6875}{288} + \frac{1375 i}{432} - \frac{9625}{31104} - \frac{1375 i}{62208} + \frac{6875}{5971968} + \frac{6875 i}{161243136} - \frac{1375}{1289945088} - \frac{125 i}{2 ^{17} \cdot 59049} + \frac{125}{1114512556032}

= \frac{3000 i + 1625}{3} - \frac{6875 i}{54} + \frac{6875}{288} + \frac{1375 i}{432} - \frac{9625}{31104} - \frac{1375 i}{62208} + \frac{6875}{5971968} + \frac{6875 i}{161243136} - \frac{1375}{1289945088} - \frac{125 i}{2 ^{17} \cdot 59049} + \frac{125}{1114512556032}

Rewrite in standard complex form:

\frac{1625 + 3000 i}{3} + (\frac{6875}{288} - \frac{9625}{31104} + \frac{6875}{5971968} - \frac{1375}{1289945088} + \frac{125}{1114512556032}) + (- \frac{6875}{54} + \frac{1375}{432} - \frac{1375}{62208} + \frac{6875}{161243136} - \frac{125}{2 ^{17} \cdot 59049}) i

= \frac{1625 + 3000 i}{3} + (\frac{6875}{288} - \frac{9625}{31104} + \frac{6875}{5971968} - \frac{1375}{1289945088} + \frac{125}{1114512556032}) + (- \frac{6875}{54} + \frac{1375}{432} - \frac{1375}{62208} + \frac{6875}{161243136} - \frac{125}{2 ^{17} \cdot 59049}) i

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