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Popolare Algebra >

espandere (1-1/2)^{21}

Soluzione

espandere

(1 - \frac{1}{2})^{21}

Soluzione

\frac{1}{2097152}

+1

Decimale

4.76837 E - 7

Fasi della soluzione

(1 - \frac{1}{2})^{21}

Applicare il teorema binomiale:

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

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

= i = 0 \sum 21 (i 21) \cdot 1^{(21 - i)} (- \frac{1}{2})^{i}

Espandere la sommatoria

= \frac{21 !}{0 ! ( 21 - 0 ) !} \cdot 1^{21} (- \frac{1}{2})^{0} + \frac{21 !}{1 ! ( 21 - 1 ) !} \cdot 1^{20} (- \frac{1}{2})^{1} + \frac{21 !}{2 ! ( 21 - 2 ) !} \cdot 1^{19} (- \frac{1}{2})^{2} + \frac{21 !}{3 ! ( 21 - 3 ) !} \cdot 1^{18} (- \frac{1}{2})^{3} + \frac{21 !}{4 ! ( 21 - 4 ) !} \cdot 1^{17} (- \frac{1}{2})^{4} + \frac{21 !}{5 ! ( 21 - 5 ) !} \cdot 1^{16} (- \frac{1}{2})^{5} + \frac{21 !}{6 ! ( 21 - 6 ) !} \cdot 1^{15} (- \frac{1}{2})^{6} + \frac{21 !}{7 ! ( 21 - 7 ) !} \cdot 1^{14} (- \frac{1}{2})^{7} + \frac{21 !}{8 ! ( 21 - 8 ) !} \cdot 1^{13} (- \frac{1}{2})^{8} + \frac{21 !}{9 ! ( 21 - 9 ) !} \cdot 1^{12} (- \frac{1}{2})^{9} + \frac{21 !}{10 ! ( 21 - 10 ) !} \cdot 1^{11} (- \frac{1}{2})^{10} + \frac{21 !}{11 ! ( 21 - 11 ) !} \cdot 1^{10} (- \frac{1}{2})^{11} + \frac{21 !}{12 ! ( 21 - 12 ) !} \cdot 1^{9} (- \frac{1}{2})^{12} + \frac{21 !}{13 ! ( 21 - 13 ) !} \cdot 1^{8} (- \frac{1}{2})^{13} + \frac{21 !}{14 ! ( 21 - 14 ) !} \cdot 1^{7} (- \frac{1}{2})^{14} + \frac{21 !}{15 ! ( 21 - 15 ) !} \cdot 1^{6} (- \frac{1}{2})^{15} + \frac{21 !}{16 ! ( 21 - 16 ) !} \cdot 1^{5} (- \frac{1}{2})^{16} + \frac{21 !}{17 ! ( 21 - 17 ) !} \cdot 1^{4} (- \frac{1}{2})^{17} + \frac{21 !}{18 ! ( 21 - 18 ) !} \cdot 1^{3} (- \frac{1}{2})^{18} + \frac{21 !}{19 ! ( 21 - 19 ) !} \cdot 1^{2} (- \frac{1}{2})^{19} + \frac{21 !}{20 ! ( 21 - 20 ) !} \cdot 1^{1} (- \frac{1}{2})^{20} + \frac{21 !}{21 ! ( 21 - 21 ) !} \cdot 1^{0} (- \frac{1}{2})^{21}

Semplificare

\frac{21 !}{0 ! ( 21 - 0 ) !} \cdot 1^{21} (- \frac{1}{2})^{0} : 1

Semplificare

\frac{21 !}{1 ! ( 21 - 1 ) !} \cdot 1^{20} (- \frac{1}{2})^{1} : - \frac{21}{2}

Semplificare

\frac{21 !}{2 ! ( 21 - 2 ) !} \cdot 1^{19} (- \frac{1}{2})^{2} : \frac{105}{2}

Semplificare

\frac{21 !}{3 ! ( 21 - 3 ) !} \cdot 1^{18} (- \frac{1}{2})^{3} : - \frac{665}{4}

Semplificare

\frac{21 !}{4 ! ( 21 - 4 ) !} \cdot 1^{17} (- \frac{1}{2})^{4} : \frac{5985}{16}

Semplificare

\frac{21 !}{5 ! ( 21 - 5 ) !} \cdot 1^{16} (- \frac{1}{2})^{5} : - \frac{20349}{32}

Semplificare

\frac{21 !}{6 ! ( 21 - 6 ) !} \cdot 1^{15} (- \frac{1}{2})^{6} : \frac{6783}{8}

Semplificare

\frac{21 !}{7 ! ( 21 - 7 ) !} \cdot 1^{14} (- \frac{1}{2})^{7} : - \frac{14535}{16}

Semplificare

\frac{21 !}{8 ! ( 21 - 8 ) !} \cdot 1^{13} (- \frac{1}{2})^{8} : \frac{101745}{128}

Semplificare

\frac{21 !}{9 ! ( 21 - 9 ) !} \cdot 1^{12} (- \frac{1}{2})^{9} : - \frac{146965}{256}

Semplificare

\frac{21 !}{10 ! ( 21 - 10 ) !} \cdot 1^{11} (- \frac{1}{2})^{10} : \frac{88179}{256}

Semplificare

\frac{21 !}{11 ! ( 21 - 11 ) !} \cdot 1^{10} (- \frac{1}{2})^{11} : - \frac{88179}{512}

Semplificare

\frac{21 !}{12 ! ( 21 - 12 ) !} \cdot 1^{9} (- \frac{1}{2})^{12} : \frac{146965}{2048}

Semplificare

\frac{21 !}{13 ! ( 21 - 13 ) !} \cdot 1^{8} (- \frac{1}{2})^{13} : - \frac{101745}{4096}

Semplificare

\frac{21 !}{14 ! ( 21 - 14 ) !} \cdot 1^{7} (- \frac{1}{2})^{14} : \frac{14535}{2048}

Semplificare

\frac{21 !}{15 ! ( 21 - 15 ) !} \cdot 1^{6} (- \frac{1}{2})^{15} : - \frac{6783}{4096}

Semplificare

\frac{21 !}{16 ! ( 21 - 16 ) !} \cdot 1^{5} (- \frac{1}{2})^{16} : \frac{20349}{65536}

Semplificare

\frac{21 !}{17 ! ( 21 - 17 ) !} \cdot 1^{4} (- \frac{1}{2})^{17} : - \frac{5985}{131072}

Semplificare

\frac{21 !}{18 ! ( 21 - 18 ) !} \cdot 1^{3} (- \frac{1}{2})^{18} : \frac{665}{131072}

Semplificare

\frac{21 !}{19 ! ( 21 - 19 ) !} \cdot 1^{2} (- \frac{1}{2})^{19} : - \frac{105}{262144}

Semplificare

\frac{21 !}{20 ! ( 21 - 20 ) !} \cdot 1^{1} (- \frac{1}{2})^{20} : \frac{21}{1048576}

Semplificare

\frac{21 !}{21 ! ( 21 - 21 ) !} \cdot 1^{0} (- \frac{1}{2})^{21} : - \frac{1}{2097152}

= 1 - \frac{21}{2} + \frac{105}{2} - \frac{665}{4} + \frac{5985}{16} - \frac{20349}{32} + \frac{6783}{8} - \frac{14535}{16} + \frac{101745}{128} - \frac{146965}{256} + \frac{88179}{256} - \frac{88179}{512} + \frac{146965}{2048} - \frac{101745}{4096} + \frac{14535}{2048} - \frac{6783}{4096} + \frac{20349}{65536} - \frac{5985}{131072} + \frac{665}{131072} - \frac{105}{262144} + \frac{21}{1048576} - \frac{1}{2097152}

1 - \frac{21}{2} + \frac{105}{2} - \frac{665}{4} + \frac{5985}{16} - \frac{20349}{32} + \frac{6783}{8} - \frac{14535}{16} + \frac{101745}{128} - \frac{146965}{256} + \frac{88179}{256} - \frac{88179}{512} + \frac{146965}{2048} - \frac{101745}{4096} + \frac{14535}{2048} - \frac{6783}{4096} + \frac{20349}{65536} - \frac{5985}{131072} + \frac{665}{131072} - \frac{105}{262144} + \frac{21}{1048576} - \frac{1}{2097152} = \frac{1}{2097152}

= \frac{1}{2097152}

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