Question

9£®ÒÑÖªÊýÁÐ{a_n}£¬{b_n}ÖУ¬a₁=1£¬b_n=£¨1-$\frac{{{a}_{n}}^{2}}{{{a}_{n+1}}^{2}}$£©•$\frac{1}{{a}_{n+1}}$£¬n¡ÊN^*£¬ÊýÁÐ{b_n}µÄÇ°nÏîºÍS_n£®
£¨1£©Èôa_n=2^n-1£¬ÇóS_n£»
£¨2£©ÊÇ·ñ´æÔڵȱÈÊýÁÐ{a_n}ʹb_n+2=S_n¶ÔÈÎÒân¡ÊN^*ºã³ÉÁ¢£¿Èô´æÔÚ£¬Çó³öËùÓÐÂú×ãÌõ¼þµÄÊýÁÐ{a_n}µÄͨÏʽ£»Èô²»´æÔÚ£¬ËµÃ÷ÀíÓÉ£®

Answer 1

·ÖÎö £¨1£©ÒÀÌâÒ⣬¿ÉÇóµÃb_n=£¨1-$\frac{{{a}_{n}}^{2}}{{{a}_{n+1}}^{2}}$£©•$\frac{1}{{a}_{n+1}}$=$\frac{3}{{2}^{n+2}}$£¬ÀûÓõȱÈÊýÁеÄÇóºÍ¹«Ê½¿ÉµÃÊýÁÐ{b_n}µÄÇ°nÏîºÍS_n£»
£¨2£©Éè´æÔÚ¸ö¹«±ÈΪqµÄµÈ±ÈÊýÁÐ{a_n}ʹb_n+2=S_n¶ÔÈÎÒân¡ÊN^*ºã³ÉÁ¢£¬ÀûÓÃn=1ʱ£¬b₃=b₁¿ÉÇóµÃq=¡À1£¬¼ìÑé¼´¿ÉµÃµ½´ð°¸£®

½â´ð ½â£º£¨1£©¡ßa₁=1£¬a_n=2^n-1£¬
¡à$\frac{1}{{a}_{n+1}}$=$\frac{1}{{2}^{n}}$£¬$\frac{{a}_{n}}{{a}_{n+1}}$=$\frac{{2}^{n-1}}{{2}^{£¨n-1£©+1}}$=$\frac{1}{2}$£¬
¡à$\frac{{{a}_{n}}^{2}}{{{a}_{n+1}}^{2}}$=$\frac{1}{4}$£¬
¡àb_n=£¨1-$\frac{{{a}_{n}}^{2}}{{{a}_{n+1}}^{2}}$£©•$\frac{1}{{a}_{n+1}}$=£¨1-$\frac{1}{4}$£©•$\frac{1}{{2}^{n}}$=$\frac{3}{{2}^{n+2}}$£¬
ÏÔÈ»£¬ÊýÁÐ{b_n}ΪÊ×ÏîΪ$\frac{3}{8}$£¬¹«±ÈΪ$\frac{1}{2}$µÄµÈ±ÈÊýÁУ¬
¡àS_n=b₁+b₂+¡­+b_n=$\frac{\frac{3}{8}[1-£¨\frac{1}{2}£©^{n}]}{1-\frac{1}{2}}$=$\frac{3}{4}$-$\frac{3}{{2}^{n+2}}$£¨n¡ÊN^*£©£®
£¨2£©Éè´æÔÚ¸ö¹«±ÈΪqµÄµÈ±ÈÊýÁÐ{a_n}ʹb_n+2=S_n¶ÔÈÎÒân¡ÊN^*ºã³ÉÁ¢£¬
Ôòn=1ʱ£¬b₁₊₂=S₁=b₁£¬¼´b₃=£¨1-$\frac{1}{{q}^{2}}$£©•$\frac{1}{{a}_{4}}$=£¨1-$\frac{1}{{q}^{2}}$£©•$\frac{1}{{a}_{2}}$£¬¼´£¨1-q²£©£¨$\frac{1}{{q}^{3}}$-$\frac{1}{q}$£©=0£¬
½âµÃ£ºq=¡À1£¬
¾­¼ìÑ飬µ±q=¡À1ʱ£¬b_n=0£¬Âú×ãb_n+2=S_n¶ÔÈÎÒân¡ÊN^*ºã³ÉÁ¢£¬
¹Ê´æÔÚ¹«±ÈΪ¡À1µÄµÈ±ÈÊýÁÐ{a_n}£¬a_n=1»ò£¨-1£©^n-1£¬Ê¹b_n+2=S_n¶ÔÈÎÒân¡ÊN^*ºã³ÉÁ¢£®

µãÆÀ ±¾Ì⿼²éÊýÁеÝÍƹØϵʽµÄÓ¦Ó㬿¼²éµÈ±ÈÊýÁеÄÐÔÖʼ°ÇóºÍ¹«Ê½µÄÓ¦Ó㬿¼²éÍÆÀíÓëÔËËãÄÜÁ¦£¬ÊôÓÚÖеµÌ⣮