Câu hỏi 23

\({3^n}.C_n^0 - {3^{n - 1}}.C_n^1 + {3^{n - 2}}.C_n^2 - {3^{n - 3}}.C_n^3 + ... + {\left( { - 1} \right)^n}.C_n^n = 2048\)  \(\left(  *  \right)\)

\( + )\)Xét khai triển: \({\left( {x - 1} \right)^n} = C_n^0.{x^n}.{\left( { - 1} \right)^0} + C_n^1.{x^{n - 1}}.{\left( { - 1} \right)^1} + C_n^2.{x^{n - 2}}.{\left( { - 1} \right)^2} + ... + C_n^n.{x^0}.{\left( { - 1} \right)^n}\)

\( + )\)Thay \(x = 3\)vào 2 vế\( \Rightarrow \)\({\left( {3 - 1} \right)^n} = C_n^0{.3^n} - C_n^1{.3^{n - 1}} + ... + C_n^n{.3^0}.{\left( { - 1} \right)^n}\)

\( \Leftrightarrow \)\({2^n} = 2048\)\( \Rightarrow \)\(n = 11\)

Chọn C