Cho số nguyên dương $n$ và $k= 1,\,2,\,3,\,...\,,n- 1$$.$ Chứng minh rằng$:$
$$a_{\,k}= \frac{2\,n^{\,2}(\,n^{\,2}- 1^{\,2}\,)(\,n^{\,2}- 2^{\,2}\,)\,...\,[\,n^{\,2}- (\,k- 1\,)^{\,2}\,]}{(\,2\,k\,)\,!}- \frac{2\,n(\,n- 1\,)(\,n- 2\,)\,...\,[\,n- (\,k- 1\,)\,]}{k\,!\,2^{\,\frac{k}{n}}}\geqq 0$$
Áp dụng$:$ $($$t\geqq 0$$)$
$$\left ( \frac{t+ \sqrt{t^{\,2}+ 4\,t}}{2}+ 1 \right )^{\,n}+ \left ( \frac{t- \sqrt{t^{\,2}+ 4\,t}}{2}+ 1 \right )^{\,n}- \left ( t+ 2^{\,\frac{1}{n}} \right )^{\,n}\geqq 0$$
