回復 25# Ling 的帖子
第3題
\(\begin{align}
& {{\left( 1+x \right)}^{n}}=C_{0}^{n}+C_{1}^{n}x+C_{2}^{n}{{x}^{2}}+C_{3}^{n}{{x}^{3}}+\cdots +C_{n}^{n}{{x}^{n}} \\
& n{{\left( 1+x \right)}^{n-1}}=C_{1}^{n}+2C_{2}^{n}x+3C_{3}^{n}{{x}^{2}}+\cdots +nC_{n}^{n}{{x}^{n-1}} \\
& nx{{\left( 1+x \right)}^{n-1}}=C_{1}^{n}x+2C_{2}^{n}{{x}^{2}}+3C_{3}^{n}{{x}^{3}}+\cdots +nC_{n}^{n}{{x}^{n}} \\
& \sum\limits_{k=1}^{n}{\left( kC_{k}^{n}\times {{3}^{k}} \right)=3n\times {{4}^{n-1}}\ge 390000} \\
& n=7,3n\times {{4}^{n-1}}=21\times {{4}^{6}}=86016<390000 \\
& n=8,3n\times {{4}^{n-1}}=24\times {{4}^{7}}=393216>390000 \\
\end{align}\)
