ÌâÄ¿ÄÚÈÝ

£¨¢ñ£©ÒÑÖªº¯Êýf(x)=
x
x+1
£®ÊýÁÐ{an}Âú×㣺an£¾0£¬a1=1£¬ÇÒ
an+1
=f(
an
)
£¬¼ÇÊýÁÐ{bn}µÄÇ°nÏîºÍΪSn£¬ÇÒSn=
2
2
[
1
an
+(
2
+1)n]
£®ÇóÊýÁÐ{bn}µÄͨÏʽ£»²¢ÅжÏb4+b6ÊÇ·ñÈÔΪÊýÁÐ{bn}ÖеÄÏÈôÊÇ£¬ÇëÖ¤Ã÷£»·ñÔò£¬ËµÃ÷ÀíÓÉ£®
£¨¢ò£©Éè{cn}ΪÊ×ÏîÊÇc1£¬¹«²îd¡Ù0µÄµÈ²îÊýÁУ¬ÇóÖ¤£º¡°ÊýÁÐ{cn}ÖÐÈÎÒⲻͬÁ½ÏîÖ®ºÍÈÔΪÊýÁÐ{cn}ÖеÄÏµÄ³äÒªÌõ¼þÊÇ¡°´æÔÚÕûÊým¡Ý-1£¬Ê¹c1=md¡±£®
·ÖÎö£º£¨¢ñ£©ÓÉÌâÒâÖª
1
an+1
-
1
an
=1
£¬
1
an
=1+(n-1)=n
£¬ËùÒÔan=
1
n2
£®ÔÙÓÉÌâÉèÌõ¼þ¿ÉÒÔµ¼³öbn=Sn-Sn-1=1+
2
 n
£¬ÓÉ´Ë¿ÉÖªb4+b6²»ÔÚÊýÁÐ{bn}ÖУ®
£¨¢ò£©ÏÈÖ¤³ä·ÖÐÔ£ºÈô´æÔÚÕûÊým¡Ý-1£¬Ê¹c1=md£®ÔÙÖ¤±ØÒªÐÔ£ºÈôÊýÁÐ{cn}ÖÐÈÎÒⲻͬÁ½ÏîÖ®ºÍÈÔΪÊýÁÐ{cn}ÖеÄÏÔòcs=c1+£¨s-1£©d£¬ct=c1+£¨t-1£©d£®
½â´ð£º½â£º£¨¢ñ£©ÒòΪ
an+1
=f(
an
)=
an
an
+1
£¬
ËùÒÔ
1
an+1
=
1
an
+1
£¬
¼´
1
an+1
-
1
an
=1
£¬
1
an
=1+(n-1)=n
£¬
¼´an=
1
n2
£®£¨4·Ö£©
ÒòΪSn=
2
2
[
1
an
+(
2
+1)n]=
2
2
n2+(1+
2
2
)n
£¬
µ±n=1ʱ£¬S1=b1=
2
+1
£¬
µ±n¡Ý2ʱ£¬bn=Sn-Sn-1=1+
2
 n
£¬
ËùÒÔbn=
2
 n+1(n¡ÊN*)
£®£¨6·Ö£©
ÓÖÒòΪb4+b6=4
2
+1+6
2
+1=10
2
+2
£¬
ËùÒÔÁîbt=10
2
+2 (t¡ÊN*)
£¬
Ôò10
2
+2=
2
t+1
£»
µÃµ½t=10+
2
2
Óët¡ÊN*ì¶Ü£¬
ËùÒÔb4+b6²»ÔÚÊýÁÐ{bn}ÖУ®£¨8·Ö£©
£¨¢ò£©³ä·ÖÐÔ£ºÈô´æÔÚÕûÊým¡Ý-1£¬Ê¹c1=md£®
Éècr£¬ctΪÊýÁÐ{cn}Öв»Í¬µÄÁ½Ï
Ôòcr+ct=c1+£¨r-1£©d+c1+£¨t-1£©d=c1+£¨r+m+t-2£©d=c1+[£¨r+m+t-1£©-1]d£®
ÓÖr+t¡Ý3ÇÒm¡Ý-1£¬ËùÒÔr+m+t-1¡Ý1£®
¼´cr+ctÊÇÊýÁÐ{cn}µÄµÚr+m+t-1Ï£¨11·Ö£©
±ØÒªÐÔ£ºÈôÊýÁÐ{cn}ÖÐÈÎÒⲻͬÁ½ÏîÖ®ºÍÈÔΪÊýÁÐ{cn}ÖеÄÏ
Ôòcs=c1+£¨s-1£©d£¬ct=c1+£¨t-1£©d£¬
£¨s£¬tΪ»¥²»ÏàͬµÄÕýÕûÊý£©
Ôòcs+ct=2c1+£¨s+t-2£©d£¬Áîcs+ct=cl£¬
µÃµ½2c1+£¨s+t-2£©d=c1+£¨l-1£©d£¨n£¬t£¬s¡ÊN*£©£¬
ËùÒÔc1=£¨l-s-t+1£©d£¬
ÁîÕûÊým=l-s-t+1£¬ËùÒÔc1=md£® £¨14·Ö£©
ÏÂÖ¤ÕûÊým¡Ý-1
ÈôÉèÕûÊým£¼-1£¬Ôò-m¡Ý2£®Áîk=-m£¬
ÓÉÌâÉèÈ¡c1£¬ckʹc1+ck=cr£¨r¡Ý1£©
¼´c1+c1+£¨k-1£©d=c1+£¨r-1£©d£¬
ËùÒÔmd+£¨-m-1£©d=£¨r-1£©d
¼´rd=0Óër¡Ý1£¬d¡Ù0Ïàì¶Ü£¬ËùÒÔm¡Ý-1£®
×ÛÉÏ£¬ÊýÁÐ{cn}ÖÐÈÎÒⲻͬÁ½ÏîÖ®ºÍÈÔΪÊýÁÐ{cn}ÖеÄÏîµÄ³äÒªÌõ¼þÊÇ´æÔÚÕûÊým¡Ý-1£¬Ê¹c1=md£®£¨16·Ö£©
µãÆÀ£º±¾Ì⿼²éÊýÁеÄÐÔÖʺÍ×ÛºÏÔËÓã¬ÄѶȽϴ󣮽âÌâʱҪÈÏÕæÉóÌ⣬×Ðϸ½â´ð£®
Á·Ï°²áϵÁдð°¸
Ïà¹ØÌâÄ¿

Î¥·¨ºÍ²»Á¼ÐÅÏ¢¾Ù±¨µç»°£º027-86699610 ¾Ù±¨ÓÊÏ䣺58377363@163.com

¾«Ó¢¼Ò½ÌÍø