每日一题[1025]数列递推与求和

已知数列$\{a_n\}$满足$a_1=1$,$a_{n+1}\cdot a_n=\dfrac 1n$($n\in\mathbb N^*$).
(1) 求证:$\dfrac{a_{n+2}}{n}=\dfrac{a_n}{n+1}$;
(2) 求证:$2\left(\sqrt{n+1}-1\right)\leqslant \dfrac{1}{2a_3}+\dfrac{1}{3a_4}+\cdots+\dfrac{1}{(n+1)a_{n+2}}\leqslant n$.


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分析与解 (1) 根据题意,有\[\begin{split} \dfrac{a_{n+2}}{n}=&\dfrac{\dfrac{1}{n+1}\cdot \dfrac{1}{a_{n+1}}}{n}\\=&\dfrac{na_n}{n(n+1)}=\dfrac{a_n}{n+1}.\end{split}\]
(2) 根据第$(1)$小题的结论,有\[\dfrac{1}{2a_3}+\dfrac{1}{3a_4}+\cdots+\dfrac{1}{(n+1)a_{n+2}}=a_2+a_3+\cdots+a_{n+1}.\]
右边不等式 根据第$(1)$小题的结论,有\[\dfrac{a_{n+2}}{a_n}=\dfrac{n}{n+1}<1,\]于是数列的奇子列和偶子列均单调递减,结合$a_1=a_2=1$,可得$a_n\leqslant 1,n\in\mathbb N^*$,于是右边不等式得证.

左边不等式 由于\[\begin{split}\dfrac{1}{a_n\cdot a_{n+1}}&=n,\\ \dfrac{1}{a_{n+1}\cdot a_{n+2}}&=n+1,\end{split}\]于是\[\dfrac{1}{a_{n+1}}\left(\dfrac{1}{a_{n+2}}-\dfrac{1}{a_n}\right)=1,\]从而\[a_{n+1}=\dfrac{1}{a_{n+2}}-\dfrac{1}{a_n}.\]因此\[\begin{split} a_2+a_3+\cdots+a_{n+1}=&\dfrac{1}{a_{n+2}}+\dfrac{1}{a_{n+1}}-\dfrac{1}{a_1}-\dfrac{1}{a_2}\\\geqslant &\dfrac{2}{\sqrt{a_{n+1}a_{n+2}}}-2\\=&2\left(\sqrt{n+1}-1\right),\end{split} \]于是左边不等式得证.

综上所述,原命题得证.

 本题来自尬题19.

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