Question

ÒÑÖª£¬ÊýÁÐ{a_n}ÓÐa₁=a£¬a₂=p£¨³£Êýp£¾0£©£¬¶ÔÈÎÒâµÄÕýÕûÊýn£¬S_n=a₁+a₂+¡­+a_n£¬²¢ÓÐS_nÂú×ãSn=

n(an-a1)

2

£®
£¨1£©ÇóaµÄÖµ£»
£¨2£©ÊÔÈ·¶¨ÊýÁÐ{a_n}ÊDz»ÊǵȲîÊýÁУ¬ÈôÊÇ£¬Çó³öÆäͨÏʽ£®Èô²»ÊÇ£¬ËµÃ÷ÀíÓÉ£»
£¨3£©Áîpn=

Sn+2

Sn+1

+

Sn+1

Sn+2

£¬ÊÇ·ñ´æÔÚÕýÕûÊýM£¬Ê¹²»µÈʽp₁+p₂+¡­+p_n-2n¡ÜMºã³ÉÁ¢£¬Èô´æÔÚ£¬Çó³öMµÄ×îСֵ£¬Èô²»´æÔÚ£¬ËµÃ÷ÀíÓÉ£®

Answer 1

·ÖÎö£º£¨1£©ÓÉ a=a₁=s₁ ºÍ Sn=

n(an-a1)

2

 ¿ÉµÃ a µÄÖµ£®
£¨2£©ÏÈÇó³ö S_n£¬¿ÉµÃ S_n-1£¬¸ù¾ÝS_n-S_n-1=a_n£¬»¯¼ò¿ÉµÃ 

an

an-1

=

n-1

n-2

£¬a_n =k£¨n-1£©£¬¹ÊÊýÁÐ{a_n}ÊǵȲîÊýÁУ®ÓÉa₂ =p=k•£¨2-1£©£¬Çó³ö k Öµ£¬µÃµ½a_n =p£¨n-1£©=£¨n-1£©p£®
£¨3£©¸ù¾Ý¶¨ÒåÏȱíʾ³öp₁+p₂+¡­+p_n-2n=2+1-

2

n+1

-

2

n+2

£¬ÔÙÇóÆäÉϱ߽缴¿É£®

½â´ð£º½â£º£¨1£©ÓÉÒÑÖª£¬µÃs1=

1•(a-a)

2

=a1=a£¬¡àa=0
£¨2£©ÓÉa₁=0µÃSn=

nan

2

£¬ÔòSn+1=

(n+1)an+1

2

£¬
¡à2£¨S_n+1-S_n£©=£¨n+1£©a_n+1-na_n£¬¼´2a_n+1=£¨n+1£©a_n+1-na_n£¬
ÓÚÊÇÓУ¨n-1£©a_n+1=na_n£¬²¢ÇÒÓÐna_n+2=£¨n+1£©a_n+1£¬
¡àna_n+2-£¨n-1£©a_n+1=£¨n+1£©a_n+1-na_n£¬¼´n£¨a_n+2-a_n+1£©=n£¨a_n+1-a_n£©£¬
¶ønÊÇÕýÕûÊý£¬Ôò¶ÔÈÎÒân¡ÊN¶¼ÓÐa_n+2-a_n+1=a_n+1-a_n£¬
¡àÊýÁÐ{a_n}ÊǵȲîÊýÁУ¬ÆäͨÏʽÊÇa_n=£¨n-1£©p£®
£¨3£©¡ßSn=

n(n-1)p

2

¡àpn=

(n+2)(n+1)p 2

(n+1)np 2

+

(n+1)np 2

(n+2)(n+1)p 2

=2+

2

n

-

2

n+2

¡àp₁+p₂+p₃+¡­+p_n-2n=(2+

2

1

-

2

3

)+(2+

2

2

-

2

4

)+¡­+(2+

2

n

-

2

n+2

)-2n
=2+1-

2

n+1

-

2

n+2

£»
ÓÉnÊÇÕýÕûÊý¿ÉµÃp₁+p₂+¡­+p_n-2n£¼3£¬
¹Ê´æÔÚ×îСµÄÕýÕûÊýM=3£¬Ê¹²»µÈʽp₁+p₂+¡­+p_n-2n¡ÜMºã³ÉÁ¢£®

µãÆÀ£º±¾Ì⿼²éÊýÁеÄ×ÛºÏÎÊÌ⣬¿¼²éÊýÁеĵÝÍƹØϵÓëͨÏʽ֮¼äµÄ¹Øϵ£¬¿¼²éѧÉú̽¾¿ÐÔÎÊÌâµÄ½â¾ö·½·¨£¬×¢ÒâÌåÏÖת»¯Ó뻯¹é˼ÏëµÄÔËÓã®