Question

9£®ÕýÏîÊýÁÐ{a_n}Âú×㣺a₁=2£¬a₂=1£¬ÇÒ$\frac{{a}_{n-1}-{a}_{n}}{{a}_{n}{a}_{n-1}}$=$\frac{{a}_{n}-{a}_{n+1}}{{a}_{n}{a}_{n+1}}$£¨n¡Ý2£©£¬Ôò´ËÊýÁеĵÚ2 016ÏîΪ£¨¡¡¡¡£©

• A£® $\frac{1}{{2}^{2015}}$
• B£® $\frac{1}{{2}^{2016}}$
• C£® $\frac{1}{2016}$
• D£® $\frac{1}{1008}$

Answer 1

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½â´ð ½â£ºÓÉ$\frac{{a}_{n-1}-{a}_{n}}{{a}_{n}{a}_{n-1}}$=$\frac{{a}_{n}-{a}_{n+1}}{{a}_{n}{a}_{n+1}}$£¨n¡Ý2£©£¬¿ÉÖª£º$\frac{1}{{a}_{n}}$-$\frac{1}{{a}_{n-1}}$=$\frac{1}{{a}_{n+1}}$-$\frac{1}{{a}_{n}}$£¬
¹ÊÊýÁÐ$\{\frac{1}{{a}_{n}}\}$ΪµÈ²îÊýÁУ¬ÓÚÊÇ$\frac{1}{{a}_{n}}$=$\frac{1}{2}$+£¨n-1£©¡Á$\frac{1}{2}$=$\frac{n}{2}$£¬
ËùÒÔa_n=$\frac{2}{n}$£¬ÓÚÊÇa₂₀₁₆=$\frac{1}{1008}$£¬
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