·¢²¼Ê±¼ä : ÐÇÆÚÒ» ÎÄÕÂ2021½ì¿Î±ê°æ¸ß¿¼ÎÄ¿ÆÊýѧһÂÖ¸´Ï°½ÌʦÓÃÊ飺µÚÁùÕµÚËĽ² ÊýÁÐÇóºÍ¼°ÊýÁеÄ×ÛºÏÓ¦ÓøüÐÂÍê±Ï¿ªÊ¼ÔĶÁb2e70e06bf64783e0912a21614791711cc797925
µÚËĽ² ÊýÁÐÇóºÍ¼°ÊýÁеÄ×ÛºÏÓ¦ÓÃ
¿¼·¨1 ÊýÁÐÇóºÍ
ÃüÌâ½Ç¶È1 Óù«Ê½·¨ºÍ·Ö×éת»¯·¨ÇóºÍ
1 [2019ɽ¶«ÎåµØÁª¿¼]ÒÑÖªµÈ²îÊýÁÐ{an}µÄÇ°nÏîºÍΪSn,ÇÒ¹ØÓÚxµÄ²»µÈʽa1x2 - S2x+2<0µÄ½â¼¯Îª(1,2). (1)ÇóÊýÁÐ{an}µÄͨÏʽ;
(2)ÈôÊýÁÐ{bn}Âú×ãbn=a2n+2???? - 1,ÇóÊýÁÐ{bn}µÄÇ°nÏîºÍTn.
(1)ÏÈÉèµÈ²îÊýÁÐ{an}µÄ¹«²îΪd,ÔÙ¸ù¾ÝÌâÒâÇó³öa1Óëd,½ø¶ø¿ÉÇó³öÊýÁÐ{an}µÄͨÏî
¹«Ê½;(2)ÏÈÓÉ(1)µÄ½áÂÛ¼°bn=a2n+2???? - 1Çó³öbn,ÔÙÀûÓõȲîÊýÁÐÓëµÈ±ÈÊýÁеÄÇóºÍ¹«Ê½,ÒÔ¼°·Ö×éת»¯·¨,¼´¿ÉÇó³ö½á¹û.
(1)ÉèµÈ²îÊýÁÐ{an}µÄ¹«²îΪd,
ÒòΪ¹ØÓÚxµÄ²»µÈʽa1x2 - S2x+2<0µÄ½â¼¯Îª(1,2),
ËùÒÔ??2=1+2=3,ÓÖS2=2a1+d,ËùÒÔa1=d, ...................................... (1,2ΪһԪ¶þ´Î·½³Ìa1x2 - S2x+2=0µÄÁ½¸ù)
1
??
Ò×Öª=2,ËùÒÔa1=1,d=1.
??1
2
ËùÒÔÊýÁÐ{an}µÄͨÏʽΪan=n. (2)ÓÉ(1)¿ÉµÃ,a2n=2n,2????=2n. ÒòΪbn=a2n+2???? - 1, ËùÒÔbn=2n - 1+2n,
ËùÒÔÊýÁÐ{bn}µÄÇ°nÏîºÍTn=(1+3+5+¡+2n - 1)+(2+22+23+¡+2n) ............................... (·Ö³ÉÁ½×éÇóºÍ) =
??(1+2?? - 1)
2
+
2(1 - 2??)1 - 2
....................................................................... (ÓõȲî(±È)ÊýÁеÄÇóºÍ¹«Ê½Ê±×¢ÒâÏîÊý)
=n2+2n+1 - 2. ½âºó·´Ë¼
´ËÌâµÄÒ×´íµãÓÐÁ½´¦:Ò»ÊǺöÊÓÊýÁÐͨÏîµÄϱê,µ¼ÖÂËùÇóµÄ½á¹û³ö´í,ÈçÔÚÇóbnʱÒ×ÎóµÃ
a2n=n,¼´µÈºÅ×ó±ßµÄϱêÒѱä,ÓұߵĴúÊýʽû±ä,µ¼ÖÂËùµÃµÄ½á¹û³ö´í;¶þÊÇÓõȲîÊýÁлòµÈ±ÈÊýÁеÄÇ°nÏîºÍ¹«Ê½Ê±,Ū´íÏîÊý,µ¼ÖÂÇóºÍ³ö´í.
1.[2019ºþÄϳ¤É³ÑÅÀñÖÐѧģÄâ]´ºÏļ¾½ÚÊÇÁ÷¸Ð¶à·¢ÆÚ,ijµØÒ½Ôº½ü30ÌìÿÌìÈë
ÔºÖÎÁÆÁ÷¸ÐµÄÈËÊýÒÀ´Î¹¹³ÉÊýÁÐ{an},ÒÑÖªa1=1,a2=2,ÇÒÂú×ãan+2 - an=1+( - 1)n(n¡ÊN*),Ôò¸ÃÒ½Ôº½ü30ÌìÈëÔºÖÎÁÆÁ÷¸ÐµÄ×ÜÈËÊýΪ .
ÃüÌâ½Ç¶È2 ÓôíλÏà¼õ·¨ÇóºÍ
2 [2017Ìì½ò,18,13·Ö]ÒÑÖª{an}ΪµÈ²îÊýÁÐ,Ç°nÏîºÍΪSn(n¡ÊN*),{bn}ÊÇÊ×ÏîΪ2µÄµÈ±ÈÊýÁÐ,ÇÒ¹«±È´óÓÚ0,b2+b3=12,b3=a4 - 2a1,S11=11b4. (¢ñ)Çó{an}ºÍ{bn}µÄͨÏʽ;
(¢ò)ÇóÊýÁÐ{a2nb2n - 1}µÄÇ°nÏîºÍ(n¡ÊN*).
(¢ñ)ÉèµÈ²îÊýÁÐ{an}µÄ¹«²îΪd,µÈ±ÈÊýÁÐ{bn}µÄ¹«±ÈΪq(q>0). ÒòΪb2+b3=12,ËùÒÔb1(q+q2)=12. ÓÖb1=2,ËùÒÔq+q2 - 6=0, ½âµÃq=2(q= - 3ÉáÈ¥),ËùÒÔbn=2n. ÓÉb3=a4 - 2a1,S11=11b4,¿ÉµÃ{??=1,½âµÃ{1ËùÒÔan=3n - 2.
??=3,
ËùÒÔÊýÁÐ{an}µÄͨÏʽΪan=3n - 2,ÊýÁÐ{bn}µÄͨÏʽΪbn=2n. (¢ò)ÓÉ(¢ñ)Öª,a2n=6n - 2,b2n - 1=2¡Á4n - 1.
ÉèÊýÁÐ{a2nb2n - 1}µÄÇ°nÏîºÍΪTn,a2nb2n - 1=(3n - 1)¡Á4n, ¹ÊTn=2¡Á4+5¡Á42+8¡Á43+¡+(3n - 1)¡Á4n ¢Ù, 4Tn=2¡Á42+5¡Á43+¡+(3n - 4)¡Á4n+(3n - 1)¡Á4n+1 ¢Ú, ¢Ù - ¢ÚµÃ,
- 3Tn=2¡Á4+3¡Á42+3¡Á43+¡+3¡Á4n - (3n - 1)¡Á4n+1 .................................... (´íλÏà¼õʱ,×¢Òâ×îºóÒ»ÏîµÄ·ûºÅ)
8=3?? - ??1,11??1+
211¡Á10
................................................ (¹¹Ôì·½³Ì×é)
??=11¡Á24,
=
12¡Á(1 - 4??)
1 - 4
- 4 - (3n - 1)¡Á4n+1 .................................................................. (Óù«Ê½·¨ÇóºÍʱ,×¢ÒâÏîÊý)
= - (3n - 2)¡Á4n+1 - 8,
3?? - 23
ËùÒÔTn=
¡Á4n+1+3.
3?? - 23
8
¹ÊÊýÁÐ{a2nb2n - 1}µÄÇ°nÏîºÍΪ
¡Á4n+1+.
3
8
2.[2020ËÄ´¨ÎåУÁª¿¼]ÉèÊýÁÐ{an}ÊǵȲîÊýÁÐ,ÊýÁÐ{bn}µÄÇ°nÏîºÍSnÂú×ã
2Sn=3(bn - 1)ÇÒa1=b1,a4=b2. (1)ÇóÊýÁÐ{an}ºÍ{bn}µÄͨÏʽ; (2)Çó{anbn}µÄÇ°nÏîºÍTn.
ÃüÌâ½Ç¶È3 ÓÃÁÑÏîÏàÏû·¨ÇóºÍ
3[2019¹ã¶«»ÝÖݵÚÈý´Îµ÷ÑÐ]ÒÑÖªÊýÁÐ{an}µÄÇ°nÏîºÍSnÂú×ã2Sn=n2+3n,n¡ÊN*. (1)Çó{an}µÄͨÏʽ; (2)ÇóÊýÁÐ{??
1
2?? - 1??2??+1
}µÄÇ°nÏîºÍTn.
??1,??=1,
¿ÉÇó³ö{an}µÄͨÏʽ;(2)ÀûÓÃ(1)µÄ½áÂÛ,Çó³öÊýÁÐ
???? - ???? - 1,??¡Ý2
(1)ÀûÓÃan={
{
1
??2?? - 1??2??+1
}µÄͨÏî,ÔÙÀûÓÃÁÑÏîÏàÏû·¨Çó³öÆäÇ°nÏîºÍTn.
(1)µ±n¡Ý2ʱ,2Sn - 1=(n - 1)2+3(n - 1), ÓÖ2Sn=n2+3n,Á½Ê½Ïà¼õµÃ2an=2n+2,ËùÒÔan=n+1. µ±n=1ʱ,2S1=2a1=4,½âµÃa1=2.
ÒòΪa1=2Âú×ãʽ×Óan=n+1, ...................................................................................... (ÑéÖ¤Ê×Ïî²»ÄÜ©) ¹Ê{an}µÄͨÏʽΪan=n+1(n¡ÊN*). (2)ÓÉ(1)Öª??
1
1
2?? - 1??2??+1
=2??(2??+2)=4¡Á??(??+1)=4(?????+1), .................................. (ÁÑÏî,×¢ÒâϵÊý4²»ÒªÂ©)
1
1
1
1111111
ËùÒÔTn=4[(1?2)+(2?3)+¡+(?????+1)] =4(1 - ??+1)
1
1
111
=
??4??+4
.
3.[2017È«¹ú¾í¢ò,15,5·Ö]µÈ²îÊýÁÐ{an}µÄÇ°nÏîºÍΪSn,a3=3,S4=10,Ôò
= .
ÃüÌâ½Ç¶È4 Óõ¹ÐòÏà¼Ó·¨ÇóºÍ
4ÒÑÖªº¯Êýf (x)=4??+2,ÊýÁÐ{an}µÄͨÏʽΪan=f (2 020),ÔòÊýÁÐ{an}µÄÇ°2 019ÏîºÍΪ .
ÓÉÌâÒâ¿ÉµÃf (x)+f (1 - x)=4??+2+41 - ??+2=4??+2+1+2¡¤4?? - 1=4??+2+2+4??=1. ÒòΪÊýÁÐ{an}µÄÇ°2 019ÏîºÍ
S2 019=f (2 020)+f (2 020)+¡+f (2 020)+f (2 020) ¢Ù, .................................... (Õâ¸öµÈʽµÄÓÒ±ßÊÇ2 019ÏîµÄºÍ) ËùÒÔS2 019=f (2 020)+f (2 020)+¡+f (2 020)+f (2 020) ¢Ú, ......................................................... (µ¹¹ýÀ´Ð´) ¢Ù+¢ÚµÃ,2S2 019=[f (ËùÒÔS2 019=
2 0192
12 020
2 019
2 018
2
1
1
2
2 018
2 0194??
41 - ??
4??
1
4??
2
4??
??
)+f (
2 0192 020
)]+[f (
2
2 020
)+f (
2 0182 020
)]+¡+[f (.
2 0192 020
)+f (
1
2 020
)]=2 019¡Á1=2 019,
,ËùÒÔÊýÁÐ{an}µÄÇ°2 019ÏîºÍΪ
2 0192
4.ÒÑ֪ƽÃæÏòÁ¿a=(lg x,1),b=(1,lg y)Âú×ãa¡¤b=12,ÇÒS=lg xn+lg(xn - 1y)+lg(xn -
22
y)+¡+lg(xyn - 1)+lg yn,ÔòS= .
ÃüÌâ½Ç¶È5 Óò¢ÏîÇóºÍ·¨ÇóºÍ
5 [2016Ìì½ò,18,13·Ö][ÎÄ]ÒÑÖª{an}ÊǵȱÈÊýÁÐ,Ç°nÏîºÍΪSn(n¡ÊN*),ÇÒ?
??1
2??3
11??2
=
,S6=63.
(1)Çó{an}µÄͨÏʽ;
2
(2)Èô¶ÔÈÎÒâµÄn¡ÊN*,bnÊÇlog2anºÍlog2an+1µÄµÈ²îÖÐÏî,ÇóÊýÁÐ{( - 1)n????}µÄÇ°2nÏîºÍ.
(1)¸ù¾ÝÒÑÖªµÈʽ¼°S6=63ÇóµÃq,½ø¶øÇóµÃÊ×Ïî,¼´¿ÉµÃµ½{an}µÄͨÏʽ;(2)ÏÈÓÉ(1)
2µÃ{bn}µÄͨÏʽ,ÔÙÓò¢ÏîÇóºÍ·¨ÇóÊýÁÐ{( - 1)n????}µÄÇ°2nÏîºÍ.
(1)ÉèÊýÁÐ{an}µÄ¹«±ÈΪq.ÓÉÒÑÖª,ÓÐ???????= ??
1
1
112
1
,½âµÃq=2»òq= - 1.ÓÖÓÉS6=a1¡¤1 - ??=63,Öª??2
1 - ??6
q¡Ù - 1,ËùÒÔa1¡¤1 - 2=63,½âµÃa1=1.ËùÒÔan=2n - 1.
(2)ÓÉÌâÒâ,µÃbn=2(log2an+log2an+1)=2(log22n - 1+log22n)=n - 2,¼´{bn}ÊÇÊ×ÏîΪ2,¹«²îΪ1µÄµÈ²îÊýÁÐ.
1
1
1
1
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