Question

ÒÑÖªº¯Êýf(x)=

0 (x¡Ü0) n[x-(n-1)]+f(n-1) (n-1£¼x¡Ün£¬n¡ÊN*)

ÊýÁÐ{a_n}Âú×ãa_n=f£¨n£©£¨n¡ÊN^*£©
£¨1£©ÇóÊýÁÐ{a_n}µÄͨÏʽ£»
£¨2£©ÉèxÖá¡¢Ö±Ïßx=aÓ뺯Êýy=f£¨x£©µÄͼÏóËùΧ³ÉµÄ·â±ÕͼÐεÄÃæ»ýΪS£¨a£©£¨a¡Ý0£©£¬ÇóS£¨n£©-S£¨n-1£©£¨n¡ÊN^*£©£»
£¨3£©ÔÚ¼¯ºÏM={N|N=2k£¬k¡ÊZ£¬ÇÒ1000¡Ük£¼1500}ÖУ¬ÊÇ·ñ´æÔÚÕýÕûÊýN£¬Ê¹µÃ²»µÈʽa_n-1005£¾S£¨n£©-S£¨n-1£©¶ÔÒ»ÇÐn£¾Nºã³ÉÁ¢£¿Èô´æÔÚ£¬ÔòÕâÑùµÄÕýÕûÊýN¹²ÓжàÉÙ¸ö£¿²¢Çó³öÂú×ãÌõ¼þµÄ×îСµÄÕýÕûÊýN£»Èô²»´æÔÚ£¬Çë˵Ã÷ÀíÓÉ£®

Answer 1

·ÖÎö£º£¨1£©ÀûÓÃÌõ¼þ£¬¿ÉµÃf£¨n£©-f£¨n-1£©=n£¬ÀûÓõþ¼Ó·¨¿ÉµÃ½áÂÛ£»
£¨2£©S£¨n£©-S£¨n-1£©ÎªÒ»Ö±½ÇÌÝÐΣ¨n=1ʱΪֱ½ÇÈý½ÇÐΣ©µÄÃæ»ý£¬¸ÃÌÝÐεÄÁ½µ×±ßµÄ³¤·Ö±ðΪf£¨n-1£©£¬f£¨n£©£¬¸ßΪ1£¬¿ÉµÃ½áÂÛ£»
£¨3£©ÉèÂú×ãÌõ¼þµÄÕýÕûÊýN´æÔÚ£¬¿ÉµÃNÖеÄÊý³ÉÊ×ÏîΪ2010£¬¹«²îΪ2µÄµÈ²îÊýÁУ¬´Ó¶ø¿ÉµÃ½áÂÛ£®

½â´ð£º½â£º£¨1£©ÓÉÌâÒ⣬f£¨n£©=n[n-£¨n-1£©]+f£¨n-1£©£¬¡àf£¨n£©-f£¨n-1£©=n
¡àf£¨1£©-f£¨0£©=1£¬f£¨2£©-f£¨1£©=2£¬¡­£¬f£¨n£©-f£¨n-1£©=n£¬
µþ¼Ó¿ÉµÃf£¨n£©-f£¨0£©=1+2+¡­+n=

n(n+1)

2

¡ßf£¨0£©=0
¡àf£¨n£©=

n(n+1)

2

¡àa_n=

n(n+1)

2

£»
£¨2£©S£¨n£©-S£¨n-1£©ÎªÒ»Ö±½ÇÌÝÐΣ¨n=1ʱΪֱ½ÇÈý½ÇÐΣ©µÄÃæ»ý£¬¸ÃÌÝÐεÄÁ½µ×±ßµÄ³¤·Ö±ðΪf£¨n-1£©£¬f£¨n£©£¬¸ßΪ1
¡àS(n)-S(n-1)=

f(n-1)+f(n)

2

¡Á1=

an-1+an

2

=

1

2

[

n(n-1)

2

+

n(n+1)

2

]=

n2

2

¡­£¨8·Ö£©
£¨3£©ÉèÂú×ãÌõ¼þµÄÕýÕûÊýN´æÔÚ£¬Ôò 

n(n+1)

2

-1005£¾

n2

2

?

n

2

£¾1005?n£¾2010
ÓÖM={2000£¬2002£¬¡­£¬2008£¬2010£¬2012£¬¡­£¬2998}
¡àN=2010£¬2012£¬¡­£¬2998¾ùÂú×ãÌõ¼þ£¬ËüÃǹ¹³ÉÊ×ÏîΪ2010£¬¹«²îΪ2µÄµÈ²îÊýÁУ®
Éè¹²ÓÐm¸öÂú×ãÌõ¼þµÄÕýÕûÊýN£¬Ôò2010+2£¨m-1£©=2998£¬½âµÃm=495
¡àMÖÐÂú×ãÌõ¼þµÄÕýÕûÊýN´æÔÚ£¬¹²ÓÐ495¸ö£¬N_min=2010¡­£¨12·Ö£©

µãÆÀ£º±¾Ì⿼²éÊýÁÐÓ뺯ÊýµÄ½áºÏ£¬¿¼²éµÈ²îÊýÁеÄͨÏ¿¼²éѧÉú·ÖÎö½â¾öÎÊÌâµÄÄÜÁ¦£¬ÊôÓÚÖеµÌ⣮