Question

9£®ÒÑÖªÊýÁÐ{a_n}£º$\frac{1}{2}$£¬$\frac{1}{3}$+$\frac{2}{3}$£¬$\frac{1}{4}+\frac{2}{4}+\frac{3}{4}$£¬¡­£¬$\frac{1}{10}+\frac{2}{10}+\frac{3}{10}+¡­+\frac{9}{10}$£¬¡­£¬Èôb_n=$\frac{1}{{a}_{n}{a}_{n+2}}$£¬ÄÇôÊýÁÐ{b_n}µÄÇ°nÏîºÍS_nµÈÓÚ£¨¡¡¡¡£©

• A£® 2-$\frac{2}{n+2}$
• B£® 3-$\frac{4n+6}{{n}^{2}+3n+2}$
• C£® $\frac{3}{2}-\frac{2n+3}{{n}^{2}+3n+2}$
• D£® 4-$\frac{4}{n+2}$

Answer 1

·ÖÎö ÓÉÊýÁÐ{a_n}£º$\frac{1}{2}$£¬$\frac{1}{3}$+$\frac{2}{3}$£¬$\frac{1}{4}+\frac{2}{4}+\frac{3}{4}$£¬¡­£¬$\frac{1}{10}+\frac{2}{10}+\frac{3}{10}+¡­+\frac{9}{10}$£¬¡­£¬¿ÉµÃ{a_n}µÄͨÏ¼´¿ÉÇó{b_n}µÄͨÏÀûÓÃÁÑÏîÏàÏû¿ÉµÃ{b_n}µÄÇ°nÏîºÍS_n£®

½â´ð ½â£ºÊýÁÐ{a_n}£º$\frac{1}{2}$£¬$\frac{1}{3}$+$\frac{2}{3}$£¬$\frac{1}{4}+\frac{2}{4}+\frac{3}{4}$£¬¡­£¬$\frac{1}{10}+\frac{2}{10}+\frac{3}{10}+¡­+\frac{9}{10}$£¬¡­£¬
¿ÉµÃ{a_n}µÄͨÏîa_n=$\frac{1+2+3+¡­n}{n+1}=\frac{n}{2}$£®
¡ßb_n=$\frac{1}{{a}_{n}{a}_{n+2}}$£¬
¡à${b}_{n}=\frac{1}{\frac{n}{2}•\frac{n+2}{2}}$=$\frac{4}{n£¨n+2£©}$=2£¨$\frac{1}{n}-\frac{1}{n+2}$£©£®
¡àÊýÁÐ{b_n}µÄÇ°nÏîºÍS_n=2£¨1-$\frac{1}{3}$+$\frac{1}{2}-$$\frac{1}{4}$$+\frac{1}{3}$-$\frac{1}{5}$+¡­+$\frac{1}{n-1}-\frac{1}{n+1}+\frac{1}{n}-\frac{1}{n+2}$£©
=2£¨$\frac{3}{2}-\frac{1}{n+1}-\frac{1}{n+2}$£©
=3-$\frac{4n+6}{{n}^{2}+3n+2}$
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