Question

£¨2012•Í¨ÖÝÇøһģ£©¶ÔÓÚÊýÁÐ{a_n}£¬´ÓµÚ¶þÏîÆð£¬Ã¿Ò»ÏîÓëËüÇ°Ò»ÏîµÄ²îÒÀ´Î×é³ÉµÈ±ÈÊýÁУ¬³Æ¸ÃµÈ±ÈÊýÁÐΪÊýÁÐ{a_n}µÄ¡°²îµÈ±ÈÊýÁС±£¬¼ÇΪÊýÁÐ{b_n}£®ÉèÊýÁÐ{b_n}µÄÊ×Ïîb₁=2£¬¹«±ÈΪq£¨qΪ³£Êý£©£®
£¨I£©Èôq=2£¬Ð´³öÒ»¸öÊýÁÐ{a_n}µÄÇ°4Ï
£¨II£©a₁ÓëqÂú×ãʲôÌõ¼þ£¬ÊýÁÐ{a_n}ÊǵȱÈÊýÁУ¬²¢Ö¤Ã÷ÄãµÄ½áÂÛ£»
£¨III£©Èôa₁=1£¬ÊýÁÐ{a_n+c_n}Êǹ«²îΪqµÄµÈ²îÊýÁУ¬ÇÒc₁=q£¬ÇóÊýÁÐ{c_n}µÄͨÏʽ£»²¢Ö¤Ã÷µ±1£¼q£¼2ʱ£¬c₅£¼-2q²£®

Answer 1

·ÖÎö£º£¨¢ñ£©¸ù¾ÝÊýÁÐ{b_n}ÊǵȱÈÊýÁУ¬ÇÒb₁=2£¬q=2£¬¿ÉµÃb₂=4£¬b₃=8£¬ÓÉ´Ë¿ÉÇó¸öÊýÁÐ{a_n}µÄÇ°4Ï
£¨¢ò£©ÏÈÈ·¶¨a₂=a₁+2£¬a₃=a₁+2+2q£¬¸ù¾ÝÊýÁÐ{a_n}ÊǵȱÈÊýÁУ¬¼´¿ÉÇó³ö¹«±È£»
£¨¢ó£©ÏÈÈ·¶¨c_n-c_n-1=q-2q^n-2£¬ÔÙÀûÓõþ¼Ó·¨£¬¼´¿ÉÇóÊýÁÐ{c_n}µÄͨÏʽ£¬´Ó¶ø¿ÉÖ¤Ã÷1£¼q£¼2ʱ£¬c₅£¼-2q²£®

½â´ð£º½â£º£¨¢ñ£©ÒòΪÊýÁÐ{b_n}ÊǵȱÈÊýÁУ¬ÇÒb₁=2£¬q=2£¬ËùÒÔb₂=4£¬b₃=8£¬
ËùÒÔa₁=1£¬a₂=3£¬a₃=1£¬a₁₅=15£®£¨Ð´³öÂú×ãÌõ¼þµÄÒ»×é¼´¿É£©¡­£¨2·Ö£©
£¨¢ò£©ÒòΪÊýÁÐ{b_n}ÊǵȱÈÊýÁУ¬Ê×Ïîb₁=2£¬¹«±ÈΪq£¬£¬ËùÒÔb₂=2q£¬b₃=2q²£¬
ËùÒÔa₂=a₁+2£¬a₃=a₁+2+2q£®
ÒòΪÊýÁÐ{a_n}ÊǵȱÈÊýÁУ¬
ËùÒÔa22=a1a3£¬¼´£¨a₁+2£©²=a₁¡Á£¨a₁+2+2q£©
ËùÒÔq=

a1+2

a1

£®
ËùÒÔµ±q=

a1+2

a1

ʱ£¬ÊýÁÐ{a_n}ÊǵȱÈÊýÁУ®¡­£¨5·Ö£©
£¨¢ó£©ÒòΪ{a_n+c_n}Êǹ«²îΪqµÄµÈ²îÊýÁУ¬ËùÒÔ£¨a_n+c_n£©-£¨a_n-1+c_n-1£©=q
¡ßa_n-a_n-1=2q^n-2£¬
¡àc_n-c_n-1=q-2q^n-2£¬
¡àc_n=c₁+£¨c₂-c₁£©+¡­+£¨c_n-c_n-1£©=nq-2£¨q^n-2+2q^n-3+¡­+q+1£©=nq-

2(1-qn-1)

1-q

¡àc₅=5q-

2(1-q4)

1-q

=-2q³-2q²+3q-2
¡àc₅-£¨-2q²£©=2q£¨1-q²£©+£¨q-2£©£®
¡ß1£¼q£¼2£¬¡à1-q²£¼0£¬q-2£¼0£®
¡àc₅-£¨-2q²£©£¼0
¡àc₅£¼-2q²£®                        ¡­£¨13·Ö£©

µãÆÀ£º±¾Ì⿼²éж¨Ò壬¿¼²éµÈ±ÈÊýÁеÄÖ¤Ã÷£¬¿¼²éµþ¼Ó·¨ÇóÊýÁеĺͣ¬ÕýÈ·Àí½âж¨ÒåÊǹؼü£®