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13.
$$\Bigg\{\; \matrix{(x-1)^3+(x-1)(2016)=-105 \cr (y-1)^3+(y-1)(2016)=105}$$，求$$x+y=$$

(建中通訊解題第53期，http://web2.ck.tp.edu.tw/~mathwe ... 30-15&Itemid=37)

110.2.11補充

[ 本帖最後由 eyeready 於 2016-6-29 09:05 PM 編輯 ]

f (3) - 4*f (5) + 6*f (7) - 4*f (9) + f (11) = 0

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$$12\times {{10}^{8}}=10\times 10\times 200\times 60000<A<20\times 20\times 300\times 70000=84\times {{10}^{8}}$$

$$A=\left( {{2}^{2}}-1 \right)\left( {{2}^{2}}+1 \right)\left( {{2}^{4}}+1 \right)\left( {{2}^{8}}+1 \right)\left( {{2}^{16}}+1 \right)={{2}^{32}}-1$$

[ 本帖最後由 thepiano 於 2016-8-2 09:42 PM 編輯 ]

\begin{align} & \sum\limits_{n=1}^{\infty }{\left\{ \frac{\sum\limits_{k=1}^{{{2}^{n}}-1}{\left[ {{\log }_{2}}k \right]}}{{{3}^{n}}} \right\}} \\ & =\frac{1\times 2}{3{}^{2}}+\frac{1\times 2+2\times 4}{3{}^{3}}+\frac{1\times 2+2\times 4+3\times 8}{3{}^{4}}+...+\frac{1\times 2+2\times 4+3\times 8+...+\left( n-1 \right)\times {{2}^{n-1}}}{3{}^{n}} \\ & =\sum\limits_{n=2}^{\infty }{\frac{\left( n-2 \right)\times {{2}^{n}}+2}{{{3}^{n}}}} \\ & =\sum\limits_{n=2}^{\infty }{\left( n-2 \right)\times {{\left( \frac{2}{3} \right)}^{n}}}+2\sum\limits_{n=2}^{\infty }{\frac{1}{{{3}^{n}}}} \\ & =\frac{8}{3}+\frac{1}{3} \\ & =3 \\ \end{align}

1*C(19,2)+2*C(18,2)+3*C(17,2)+...+18*C(2,2)=C(20,3)+C(19,3)+...+C(4,3)+C(2,2)  = C(20,4)  (巴斯卡定理)

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