试题

zhangyuong

16.$\alpha, \beta \in R$，且满足$\cos \alpha \ne \cos \beta$，那么对所有的$n \ge 2$的自然数，都有

$|\frac{\cos n\beta \cos \alpha - \cos n\alpha \cos \beta}{\cos \beta - \cos \alpha}| < n^2 -1$

zhangyuong

17.已知数列$\left\{a_n\right\}$是正项等差数列，且满足公差$d>0$，求证：

$\frac{a_1}{a_2} \cdot \frac{a_3}{a_4} \cdots \frac{a_{2n-1}}{a_{2n}} < \sqrt {\frac{a_1}{a_{2n+1}}}$

zhangyuong

18.实数$x$满足$x^3=x+1$，那么有

$x^{3n}=a_n x + b_n + \frac{c_n}{x}$

且

$
\left\{ \begin{array}{l}
a_{n+1}=a_n + b_n\\
b_{n+1}=a_n + b_n + c_n\\
c_{n+1}= a_n +c_n
\end{array} \right.
$

zhangyuong

19.设$\Phi_i$为从三角形的边$i$旋转边上对应中线所成的有向角，那么

$\sum \cos \Phi =0$