Question

ÉèÊýÁÐ{a_n}ÊÇÊ×ÏîΪ1£¬¹«²îΪ2µÄµÈ²îÊýÁУ¬¶Ôÿһ¸ök¡ÊN^*£¬ÔÚa_kÓëa_k+1Ö®¼ä²åÈë2^k-1¸ö2£¬µÃµ½ÐÂÊýÁÐ{b_n}£¬ÉèA_n¡¢B_n·Ö±ðÊÇÊýÁÐ{a_n}ºÍ{b_n}µÄÇ°nÏîºÍ£®
£¨1£©a₁₀ÊÇÊýÁÐ{b_n}µÄµÚ¼¸Ï
£¨2£©ÊÇ·ñ´æÔÚÕýÕûÊým£¬Ê¹B_m=2010£¿Èô²»´æÔÚ£¬Çë˵Ã÷ÀíÓÉ£»·ñÔò£¬Çó³ömµÄÖµ£»
£¨3£©Éèa_mÊÇÊýÁÐ{b_n}µÄµÚf£¨m£©ÏÊԱȽϣºB_f£¨m£©Óë2A_mµÄ´óС£¬ÇëÏêϸÂÛÖ¤ÄãµÄ½áÂÛ£®

Answer 1

£¨1£©ÔÚÊýÁÐ{b_n}ÖУ¬¶Ôÿһ¸öK¡ÊN^*£¬
ÔÚa_kÓëa_k+1Ö®¼äÓÐ2^k-1¸ö2£¬¡àa₁₀ÔÚÊýÁÐ{b_n}ÖеÄÏîÊýΪ£º10+1+2+4+¡­+2⁸   ¡­£¨2·Ö£©
=10+

1-29

1-2

=521¼´a10ÊÇÊýÁÐ{bn}ÖеÚ521Ïî   ¡­£¨3·Ö£©
£¨2£©a_n=1+£¨n-1£©•2=2n-1£¬ÔÚÊýÁÐ{b_n}ÖУ¬a_n¼°ÆäÇ°ÃæËùÓÐÏîµÄºÍΪ£º[1+3+5+¡­+£¨2n-1£©]+£¨2+4+¡­+2^n-1£©=n2+

2¡Á(1-2n-1)

1-2

=2n+n2-2¡­£¨5·Ö£©
¡ß2¹⁰+10²-2=1122£¼2010£¼2¹¹+11²-2
ÇÒ2010-1122=888=444¡Á2
¡à´æÔÚm=521+444=965£¬Ê¹µÃB_m=2010¡­£¨8·Ö£©
£¨3£©ÓÉ£¨2£©ÖªB_f£¨m£©=2^m+m²-2ÓÖA_m=1+3+5+¡­+£¨2m-1£©=m²
¡àB_f£¨m£©-2A_m=£¨2^m+m²-2£©-2m²=2^m-£¨m²+2£©¡­£¨10·Ö£©
µ±m=1ʱ£¬2^m=2£¬m²+2=3£¬¹Ê2^m£¼m²+2£»
µ±m=2ʱ£¬2^m=4£¬m²+2=6£¬¹Ê2^m£¼m²+2£»
µ±m=3ʱ£¬2^m=8£¬m²+2=11£¬¹Ê2^m£¼m²+2£»
µ±m=4ʱ£¬2^m=16£¬m²+2=18£¬¹Ê2^m£¼m²+2£» ¡­£¨12·Ö£©
µ±m¡Ý5ʱ£¬2m=1+

C

1m

+

C

2m

+¡­+

C

m-2m

+

C

m-1m

+1¡Ý2(1+m+

m(m-1)

2

)
Òò¶øµ±m=1£¬2£¬3£¬4ʱ£¬B_f£¨m£©£¼2A_m£»
µ±m¡Ý5ʱÇÒm¡ÊN^*ʱ£¬B_f£¨m£©£¾2A_m¡­£¨14·Ö£©