Question

20£®ÉèÊýÁÐÊýÁÐ{b_n}Âú×ãb_n+1=$\frac{1}{2}$b_n+$\frac{1}{4}$£¬ÇÒb₁=$\frac{7}{2}$£¬T_nΪ{b_n}µÄÇ°nÏîºÍ£®
£¨1£©ÇóÖ¤£ºÊýÁÐ{b_n-$\frac{1}{2}$}ÊǵȱÈÊýÁв¢ÇóÊýÁÐ{b_n}µÄͨÏʽ£»
£¨2£©Èç¹û¶ÔÈÎÒân¡ÊN^*£¬²»µÈʽ$\frac{2{T}_{n}+3•{2}^{2-n}-10}{k}$¡Ün²+4n+5Èôºã³ÉÁ¢£¬ÇóʵÊýkµÄÈ¡Öµ·¶Î§£®

Answer 1

·ÖÎö £¨1£©ÓÉb_n+1=$\frac{1}{2}$b_n+$\frac{1}{4}$£¬Á½±ßͬʱ¼õ$\frac{1}{2}$£¬ÔËÓõȱÈÊýÁеĶ¨ÒåºÍͨÏʽ£¬¼ÆËã¼´¿ÉµÃµ½ËùÇó£»
£¨2£©ÔËÓõȱÈÊýÁеÄÇóºÍ¹«Ê½£¬¿ÉµÃT_n£¬Ô­²»µÈʽ¼´Îª$\frac{1}{k}$¡Ü$\frac{{n}^{2}+4n+5}{n+2}$=£¨n+2£©+$\frac{1}{n+2}$£¬ÔËÓöԹ´º¯ÊýµÄµ¥µ÷ÐԿɵÃ×îСֵ£¬½ø¶øµÃµ½kµÄ·¶Î§£®

½â´ð ½â£º£¨1£©Ö¤Ã÷£ºÓÉb_n+1=$\frac{1}{2}$b_n+$\frac{1}{4}$£¬¿ÉµÃ
b_n+1-$\frac{1}{2}$=$\frac{1}{2}$b_n+$\frac{1}{4}$-$\frac{1}{2}$£¬¼´Îªb_n+1-$\frac{1}{2}$=$\frac{1}{2}$£¨b_n-$\frac{1}{2}$£©£¬
ÓɵȱÈÊýÁеĶ¨Òå¿ÉµÃÊýÁÐ{b_n-$\frac{1}{2}$}ÊÇÊ×ÏîΪ$\frac{7}{2}$-$\frac{1}{2}$=3£¬¹«±ÈΪ$\frac{1}{2}$µÄµÈ±ÈÊýÁУ¬
¿ÉµÃb_n-$\frac{1}{2}$=3•$\frac{1}{{2}^{n-1}}$£¬¼´Îªb_n=$\frac{1}{2}$+3•$\frac{1}{{2}^{n-1}}$£»
£¨2£©T_n=$\frac{n}{2}$+3•$\frac{1-\frac{1}{{2}^{n}}}{1-\frac{1}{2}}$=$\frac{n}{2}$+6£¨1-$\frac{1}{{2}^{n}}$£©£¬
²»µÈʽ$\frac{2{T}_{n}+3•{2}^{2-n}-10}{k}$¡Ün²+4n+5£¬¼´Îª$\frac{n+2}{k}$¡Ün²+4n+5£¬
¼´ÓÐ$\frac{1}{k}$¡Ü$\frac{{n}^{2}+4n+5}{n+2}$=£¨n+2£©+$\frac{1}{n+2}$£¬
ÓÉ£¨n+2£©+$\frac{1}{n+2}$ÔÚn¡ÊN^*µÝÔö£¬¿ÉµÃn=1ʱ£¬È¡µÃ×îСֵ$\frac{10}{3}$£®
¼´ÓÐ$\frac{1}{k}$¡Ü$\frac{10}{3}$£¬½âµÃk£¼0»òk¡Ý$\frac{3}{10}$£®
ÔòʵÊýkµÄÈ¡Öµ·¶Î§ÊÇ£¨-¡Þ£¬0£©¡È[$\frac{3}{10}$£¬+¡Þ£©£®

µãÆÀ ±¾Ì⿼²éµÈ±ÈÊýÁеĶ¨ÒåºÍͨÏʽ¡¢ÇóºÍ¹«Ê½µÄÔËÓã¬Í¬Ê±¿¼²éÊýÁв»µÈʽºã³ÉÁ¢ÎÊÌâµÄ½â·¨£¬×¢ÒâÔËÓòÎÊý·ÖÀëºÍ¶Ô¹´º¯ÊýµÄµ¥µ÷ÐÔ£¬¿¼²éÔËËãÄÜÁ¦£¬ÊôÓÚÖеµÌ⣮