Cho khai triển \({\left( {2 + 3x} \right)^{2021}} = {a_0} + {a_1}x + {a_2}{x^2}... + {a_{2021}}{x^{2021}}\). Hệ số lớn nhất trong khai triển đã cho là

Ta có: \({a_k} = C_{2021}^k{\left( {\dfrac{3}{2}} \right)^k}{.2^{2021}}\)

Giả sử \({a_k}\max \) khi đó

\(\begin{array}{l}\left\{ \begin{array}{l}{a_k} \ge {a_{k + 1}}\\{a_k} \ge {a_{k - 1}}\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}\dfrac{{2021!}}{{k!\left( {2021 - k} \right)!}}.{\left( {\dfrac{3}{2}} \right)^k} = \dfrac{{2021!}}{{\left( {k + 1} \right)!\left( {2020 - k} \right)!}}.{\left( {\dfrac{3}{2}} \right)^{k + 1}}\\\dfrac{{2021!}}{{k!\left( {2021 - k} \right)!}}.{\left( {\dfrac{3}{2}} \right)^k} = \dfrac{{2021!}}{{\left( {k - 1} \right)!\left( {2022 - k} \right)!}}.{\left( {\dfrac{3}{2}} \right)^{k - 1}}\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}\dfrac{{k + 1}}{{2021 - k}} \ge \dfrac{3}{2}\\\dfrac{{2022}}{k} \ge \dfrac{2}{3}\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}\dfrac{{\left( {k + 1} \right).2 - 3.\left( {2021 - k} \right)}}{{2.\left( {2021 - k} \right)}} \ge 0\\\dfrac{{3\left( {2022 - k} \right) - 2k}}{{3k}} \ge 0\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}1213 \le k \le 2021\\0 \le k \le 1213\end{array} \right. \Leftrightarrow k = 1213\end{array}\)