Cho 1/1 - x + 1/1 + x + 2/1 + x^2 + 4/1 + x^4 + 8/1 + x^8 = .../1 - x^16. Số thích hợp điền vào chỗ trống là A. 16 B. 8 C. 4 D. 20

Lời giải

Đáp án đúng là: A

\[\frac{1}{{1 - {\rm{x}}}} + \frac{1}{{1 + {\rm{x}}}} + \frac{2}{{1 + {{\rm{x}}^2}}} + \frac{4}{{1 + {{\rm{x}}^4}}} + \frac{8}{{1 + {{\rm{x}}^8}}}\]

\[ = \frac{{1 + x + 1 - x}}{{\left( {1 - x} \right)\left( {1 + x} \right)}} + \frac{2}{{1 + {{\rm{x}}^2}}} + \frac{4}{{1 + {{\rm{x}}^4}}} + \frac{8}{{1 + {{\rm{x}}^8}}}\]

\[ = \frac{2}{{1 - {{\rm{x}}^2}}} + \frac{2}{{1 + {{\rm{x}}^2}}} + \frac{4}{{1 + {{\rm{x}}^4}}} + \frac{8}{{1 + {{\rm{x}}^8}}}\]

\[ = \frac{{2\left( {1 + {{\rm{x}}^2}} \right) + 2\left( {1 - {{\rm{x}}^2}} \right)}}{{\left( {1 - {{\rm{x}}^2}} \right)\left( {1 + {{\rm{x}}^2}} \right)}} + \frac{4}{{1 + {{\rm{x}}^4}}} + \frac{8}{{1 + {{\rm{x}}^8}}}\]

\[ = \frac{4}{{1 - {{\rm{x}}^4}}} + \frac{4}{{1 + {{\rm{x}}^4}}} + \frac{8}{{1 + {{\rm{x}}^8}}}\]

\[ = \frac{{4\left( {1 + {{\rm{x}}^4}} \right) + 4\left( {1 - {{\rm{x}}^4}} \right)}}{{\left( {1 - {{\rm{x}}^4}} \right)\left( {1 + {{\rm{x}}^4}} \right)}} + \frac{8}{{1 + {{\rm{x}}^8}}}\]

\[ = \frac{8}{{1 - {{\rm{x}}^8}}} + \frac{8}{{1 + {{\rm{x}}^8}}} = \frac{{8\left( {1 + {{\rm{x}}^8}} \right) + 8\left( {1 - {{\rm{x}}^8}} \right)}}{{\left( {1 - {{\rm{x}}^8}} \right)\left( {1 + {{\rm{x}}^8}} \right)}} = \frac{{16}}{{1 - {x^{16}}}}\].