Question

9£®ÒÑÖªÊýÁÐ{a_n}µÄͨÏîΪa_n£¬Ç°nÏîºÍΪS_n£¬ÇÒa_nÊÇS_nÓë2µÄµÈ²îÊýÁУ¬ÊýÁÐ{b_n}ÖУ¬b₁=1£¬µãP£¨b_n£¬b_n+1£©ÔÚÖ±Ïßx-y+2=0ÉÏ£®
£¨1£©ÇóÊýÁÐ{a_n}¡¢{b_n}µÄͨÏʽ£»
£¨2£©Çó$\frac{{b}_{1}}{{a}_{1}}$+$\frac{{b}_{2}}{{a}_{2}}$+¡­+$\frac{{b}_{n}}{{a}_{n}}$µÄºÍ£»
£¨3£©ÉèÊýÁÐ{b_n}µÄÇ°nÏîºÍΪB_n£¬ÊԱȽÏ$\frac{1}{{B}_{1}}$+$\frac{1}{{B}_{2}}$+¡­+$\frac{1}{{B}_{n}}$Óë2µÄ´óС£¨·ÅËõ·¨£©

Answer 1

·ÖÎö £¨1£©ÀûÓÃÒÑÖªÌõ¼þµÃ³öÊýÁеÄͨÏîºÍÇ°nÏîºÍÖ®¼äµÄµÈʽ¹Øϵ£¬ÔÙ½áºÏ¶þÕß¼äµÄ»ù±¾¹Øϵ£¬µÃ³öÊýÁÐ{a_n}µÄͨÏʽ£¬¸ù¾Ý{b_n}µÄÏàÁÚÁ½ÏîÂú×ãµÄ¹ØϵµÃ³öµÝÍƹØϵ£¬½øÒ»²½Çó³öÆäͨÏʽ£»
£¨2£©ÀûÓôíλÏà¼õ·¨½øÐÐÇó½â¼´¿É£»
£¨3£©ÀûÓ÷ÅËõ·¨×ª»¯¸÷ÏîÊǽâ¾ö¸ÃÎÊÌâµÄ¹Ø¼ü£¬½«ËùÇóµÄ¸÷Ïî·ÅËõת»¯ÎªÄÜÇóºÍµÄÒ»¸öÊýÁеĸ÷Ïî¹À¼ÆÆäºÍ£¬½ø¶ø´ïµ½±È½Ï´óСµÄÄ¿µÄ£®

½â´ð ½â£º£¨1£©ÓÉÌâÒâ¿ÉµÃ2a_n=S_n⁺2£¬
µ±n=1ʱ£¬a₁=2£¬
µ±n¡Ý2ʱ£¬ÓÐ2a_n-1=s_n-1⁺2£¬Á½Ê½Ïà¼õ£¬ÕûÀíµÃa_n=2a_n-1£¬
¼´ÊýÁÐ{a_n}ÊÇÒÔ2ΪÊ×Ï2Ϊ¹«±ÈµÄµÈ±ÈÊýÁУ¬¹Êa_n=2ⁿ£®
ÓɵãP£¨b_n£¬b_n+1£©ÔÚÖ±Ïßx-y+2=0ÉÏ£¬µÃ³öb_n-b_n+1+2=0£¬¼´b_n+1-b_n=2£¬
¼´ÊýÁÐ{b_n}ÊÇÒÔ1ΪÊ×Ï2Ϊ¹«²îµÄµÈ²îÊýÁУ¬
Òò´Ëb_n=2n-1£»
£¨2£©ÓÉ£¨1£©¿ÉÖª$\frac{{b}_{n}}{{a}_{n}}$=$\frac{2n-1}{{2}^{n+1}}$£¬
¼ÇT_n=$\frac{{b}_{1}}{{a}_{1}}$+$\frac{{b}_{2}}{{a}_{2}}$+¡­+$\frac{{b}_{n}}{{a}_{n}}$£¬
ÔòT_n=$\frac{1}{2}$+$\frac{3}{{2}^{2}}$+$\frac{5}{{2}^{3}}$+¡­+$\frac{2n-1}{{2}^{n}}$          ¢Ù
¡à$\frac{1}{2}$Tn=$\frac{1}{{2}^{2}}$+$\frac{3}{{2}^{3}}$+$\frac{5}{{2}^{4}}$+¡­+$\frac{2n-1}{{2}^{n+1}}$       ¢Ú
¢Ù-¢ÚµÃ $\frac{1}{2}$Tn=$\frac{1}{2}$+$\frac{1}{{2}^{2}}$+$\frac{1}{{2}^{3}}$+$\frac{2}{{2}^{3}}$+¡­+$\frac{2}{{2}^{n}}$-$\frac{2n-1}{{2}^{n+1}}$£¬
¡à$\frac{{b}_{1}}{{a}_{1}}$+$\frac{{b}_{2}}{{a}_{2}}$+¡­+$\frac{{b}_{n}}{{a}_{n}}$=3-$\frac{1}{{2}^{n-2}}$-$\frac{2n-1}{{2}^{n}}$£»
£¨3£©¡ßB_n=1+3+5+¡­+£¨2n-1£©=n²
¡à$\frac{1}{{B}_{1}}$+$\frac{1}{{B}_{2}}$+¡­+$\frac{1}{{B}_{n}}$
=$\frac{1}{{1}^{2}}$+$\frac{1}{{2}^{2}}$+$\frac{1}{{3}^{2}}$+¡­+$\frac{1}{{n}^{2}}$
£¼1+$\frac{1}{1¡Á2}$+$\frac{1}{2¡Á3}$+¡­+$\frac{1}{£¨n-1£©¡Án}$
=1+1-$\frac{1}{2}$+$\frac{1}{2}$-$\frac{1}{3}$+¡­+$\frac{1}{n-1}$-$\frac{1}{n}$
=2-$\frac{1}{n}$£¼2£®

µãÆÀ ±¾Ì⿼²éµÈ²îÊýÁС¢µÈ±ÈÊýÁеÄÅж¨ÎÊÌ⣬¿¼²é¸ù¾ÝÊýÁеĵÝÍƹØϵµÃ³öÊýÁÐͨÏʽµÄ·½·¨£¬¿¼²éÊýÁеÄͨÏîÓëÇ°nÏîºÍÖ®¼äµÄ¹Øϵ£¬¿¼²éÊýÁÐÇóºÍµÄ˼ÏëºÍ·½·¨£¬¿¼²é·ÅËõ·¨¹À¼Æ²»µÈʽµÄÓйØÎÊÌ⣬¿¼²éѧÉú·ÖÎöÎÊÌâ½â¾öÎÊÌâµÄÄÜÁ¦ºÍÒâʶ£¬ÊôÓÚÖеµÌ⣮