高中数学选择性必修 第二册(381题)
已知数列 \(1,\frac{1}{2},\frac{1}{{{2^2}}},\frac{3}{{{2^2}}},\frac{1}{{{2^3}}},\frac{3}{{{2^3}}},\frac{5}{{{2^3}}}\frac{7}{{{2^3}}},...,\frac{1}{{{2^n}}},\frac{3}{{{2^n}}},\frac{5}{{{2^n}}},...\) ,则该数列第\(2019\)项是( )
A.\(\frac{{1989}}{{{2^{10}}}}\)
B.\(\frac{{2019}}{{{2^{10}}}}\)
C.\(\frac{{1989}}{{{2^{11}}}}\)
D.\(\frac{{2019}}{{{2^{11}}}}\)
知识点:第四章 数列
参考答案:C
解析:
由数列\(\left( 1 \right),\left( {\frac{1}{2}} \right),\left( {\frac{1}{{{2^2}}},\frac{3}{{{2^2}}}} \right)\left( {\frac{1}{{{2^3}}},\frac{3}{{{2^3}}},\frac{5}{{{2^3}}},\frac{7}{{{2^3}}}} \right),...,\left( {\frac{1}{{{2^n}}},\frac{3}{{{2^n}}},\frac{5}{{{2^n}}},...} \right)\),可发现其项数为
\(1,1,2,4,8,...,{2^{k - 2}},...\),则前\(12\)个括号里共有\(1024\)项,前\(13\)个括号里共有\(2048\)项,
故原数列第\(2019\)项是第\(13\)个括号里的第\(995\)项,第\(13\)个括号里的数列通项为\(\frac{{2m - 1}}{{{2^{11}}}}\),
所以第\(12\)个括号里的第\(995\)项是\(\frac{{1989}}{{{2^{11}}}}\).
故选:C.