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[toannguyenebolala]

Chứng minh rằng $C_{22}^{3}+C_{22}^{7}+C_{22}^{11}+C_{22}^{15}+C_{22}^{19}=2^{20}+2^{10}$

[Element hero Neos]

Chứng minh rằng $C_{22}^{3}+C_{22}^{7}+C_{22}^{11}+C_{22}^{15}+C_{22}^{19}=2^{20}+2^{10}$

Ta thấy có $4$ căn bậc $4$ của đơn vị là $\left\{\begin{matrix}\varepsilon_1=i\\\varepsilon_2=i^2=-1\\\varepsilon_3=i^3=-i\\\varepsilon_4=i^4=1\end{matrix}\right.$

Ta tính các khai triển sau

   $\varepsilon_1.(1+\varepsilon_1)^{22}=C_{22}^{0}.\varepsilon_1^1+C_{22}^{1}.\varepsilon_1^2+C_{22}^{2}.\varepsilon_1^3+C_{22}^{3}.\varepsilon_1^4+...+C_{22}^{22}.\varepsilon_1^{23}$

   $\varepsilon_2.(1+\varepsilon_2)^{22}=C_{22}^{0}.\varepsilon_2^1+C_{22}^{1}.\varepsilon_2^2+C_{22}^{2}.\varepsilon_2^3+C_{22}^{3}.\varepsilon_2^4+...+C_{22}^{22}.\varepsilon_2^{23}$

   $\varepsilon_3.(1+\varepsilon_3)^{22}=C_{22}^{0}.\varepsilon_3^1+C_{22}^{1}.\varepsilon_3^2+C_{22}^{2}.\varepsilon_3^3+C_{22}^{3}.\varepsilon_3^4+...+C_{22}^{22}.\varepsilon_3^{23}$

   $\varepsilon_4.(1+\varepsilon_4)^{22}=C_{22}^{0}.\varepsilon_4^1+C_{22}^{1}.\varepsilon_4^2+C_{22}^{2}.\varepsilon_4^3+C_{22}^{3}.\varepsilon_4^4+...+C_{22}^{22}.\varepsilon_3^{23}$ 

Do đó

   $\sum_{i=1}^{4}\varepsilon_i.(1+\varepsilon_i)^{22}=C_{22}^{0}(\varepsilon_1^1+\varepsilon_2^1+\varepsilon_3^1+\varepsilon_4^1)+C_{22}^{1}(\varepsilon_1^2+\varepsilon_2^2+\varepsilon_3^2+\varepsilon_4^2)+C_{22}^{2}(\varepsilon_1^3+\varepsilon_2^3+\varepsilon_3^3+\varepsilon_4^3)+C_{22}^{3}(\varepsilon_1^4+\varepsilon_2^4+\varepsilon_3^4+\varepsilon_4^4)+...+C_{22}^{22}(\varepsilon_1^{23}+\varepsilon_2^{23}+\varepsilon_3^{23}+\varepsilon_4^{23})=4(C_{22}^{3}+C_{22}^{7}+C_{22}^{11}+C_{22}^{15}+C_{22}^{19})$

Mặt khác ta có

   $\sum_{i=1}^{4}\varepsilon_i.(1+\varepsilon_i)^{22}=\varepsilon_1.(1+\varepsilon_1)^{22}+\varepsilon_2.(1+\varepsilon_2)^{22}+\varepsilon_3.(1+\varepsilon_3)^{22}+\varepsilon_4.(1+\varepsilon_4)^{22}=i.(-2^{11})i+(-1).0+(-i).2^{11}i+2^{22}=2^{22}+2^{12}$

Vậy ta có đpcm.

Bài viết đã được chỉnh sửa nội dung bởi Element hero Neos: 29-10-2017 - 16:22