重难点17新情景、新定义下的数列问题七大题型（原卷版）_11.4-2_2026-11-04

      
      
重难点17 新情景、新定义下的数列问题【七大题型】

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cdefgh
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 -a(‡ˆ|a|);‰[…tu 014a„Z[+†.
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Š‹./';†ŒŽ;./ ‘8;’“./ 01;”./ 
or;“–—˜™”;›œ;#$;ž;Ÿ ' (s¡.
(2)%&'
,y¢£@¤ 
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º‹pqs ¯ .
 1  
 !"
» 1¼2024·½¾¿À·Á§ÂY=
{𝑎𝑛};}.Δ𝑎𝑛~
{𝑎𝑛} @Ãď;ÅΔ𝑎𝑛= 𝑎𝑛+1−𝑎𝑛
(𝑛∈N*);}.Δ𝑘𝑎𝑛~
{𝑎𝑛} Ã𝑘ď;ÅΔ𝑘𝑎𝑛= Δ𝑘−1𝑎𝑛+1−Δ𝑘−1𝑎𝑛(𝑛∈N*).Æ𝑎𝑛=
𝑛(𝑛−1)(2𝑛−1)
6
;Ç
      
      
Δ2𝑎3 = ¼    Â
Av7
Bv9
Cv11
Dv13
Èa 1-1¼2024·ÉÊËÌ·Á§ÂÍ1,2, ⋅⋅⋅,𝑛”–ΦÏÐÑÒ@ ‹
{𝑎𝑛};YÓÔ1 ≤𝑖< 𝑗≤𝑛;
ÕÖ𝑎𝑖> 𝑎𝑗;×ØÙ
Y(𝑎𝑖,𝑎𝑗)^Ò
{𝑎𝑛} @¤ÚÐY.Æ𝑛= 4;ÇÛc 2 ¤ÚÐY 
{𝑎𝑛} ¤
~
¼    Â
Av4
Bv5
Cv6
Dv7
Èa 1-2¼23-24 Á¬·ÜÝ[Þ·ÃßàáÂ
âãä×åNæçèéêë'd;TUcìí@¤

{𝑎𝑛}
1;1;2;3;5;8…;Åîï 3 *ð;ñ@*ò|=Xóôõ*?);ö𝑎1 = 𝑎2 = 1;𝑎𝑛+2 =
𝑎𝑛+1 + 𝑎𝑛;ìí 
Ù~“ãä×å
”.Æ𝑎𝑚= 2(𝑎3 + 𝑎6 + 𝑎9 + ⋯+ 𝑎174) +1;Ç𝑚= ¼    Â
Av175
Bv176
Cv177
Dv178
Èa 1-3¼2024·EF·÷Â0-1 øùÐN,úûchüo†.ÆÐ𝑎1𝑎2⋯𝑎𝑛⋯ýþ𝑎𝑖
∈{0,1}(𝑖= 1,2,⋯);ÿĀNāĂ
𝑚;Ÿ 𝑎𝑖+𝑚= 𝑎𝑖(𝑖= 1,2,⋯)Òă;ÇÙÅ~ 0-1 øùÐ;ĄÙýþ
𝑎𝑖+𝑚= 𝑎𝑖(𝑖= 1,2,⋯) ąĆāĂ
𝑚~ì¤Ð øù.Y=øù~𝑚 0-1 Ð𝑎1𝑎2⋯𝑎𝑛⋯;𝐶(𝑘) =
1
𝑚
𝑚
𝑖=1𝑎𝑖𝑎𝑖+𝑘(𝑘= 1,2,⋯,𝑚−1)6ćĈÅ01 üoĉĊ;5øù~ 5 0-1 Ð;ýþ𝐶(𝑘) ≤
1
5
(𝑘= 1,2,3,4) Ð6¼    Â
Av11010⋯
Bv11011⋯
Cv10001⋯
Dv11001⋯
 2  
 #$
» 2¼2024·ċ¿·§ČčĎ“ďĐđĒ”ēĔĕĖ=ėF;ˆĘęĖ‹Ěě;Ĝ„@RĝĞďĐğĠ 
ġĢģĤĥXĦ‹ħě;ĨĩģKĪÏīĥXĞĬ“ďĐđĒ”.“ďĐđĒ”6ĉ@¤āĂ
;ÕÖ6ĭ
Ī
Į( 3 įİ 1;ÕÖ6ı
ĪIJ( 2;ìíijyÆĴĵ#$;ąĶķ‹ 1.YÓÔāĂ
𝑎0;”–¬Ĉ}Ç·
ĸï𝑛ĵ#$ ‰Ö~𝑎𝑛(𝑛∈N);Æ𝑎5 = 1;ÿ𝑎𝑖(𝑖= 1,2,3,4)ň~ 1;Ç𝑎0 = ¼    Â
Av5 ‡ 16
Bv5 ‡ 32
Cv5 ‡ 16 ‡ 4
Dv5 ‡ 32 ‡ 4
Èa 2-1¼2023·ÊĺĻļ·@§Â
{𝑎𝑛};𝑎𝑛= log𝑛+1(𝑛+ 2) (𝑛∈N∗);./
Ÿ𝑎1 ⋅𝑎2 ⋅  ⋯   ⋅𝑎𝑘~
Ă
 
𝑘  (𝑘∈N∗)ĞĽùľ
;ÇĿŀ[1,2023]9 ´cùľ
 )|=¼    Â
Av2023
Bv2024
Cv2025
Dv2026
Èa 2-2¼2024·¬ŁłŃ·ń§ÂÍāĂ
𝑛s~õ¤āĂ
𝑘14𝑘2 Ņ;ö𝑛= 𝑘1 ⋅𝑘2;ņ𝑘14𝑘2õ

Ä ŇYňąĆd;JKÙÅ~ąʼnsvÕ20 = 1 × 20 = 2 × 10 = 4 × 5;Å 4×5 ö~ 20 ąʼn
      
      
s;ņ𝑘14𝑘26𝑛 ąʼnsd;./𝑓(𝑛) = |𝑘1−𝑘2|;Ç
{𝑓(5𝑛)} ó 2023 * )~¼    Â
Av51012
Bv51012−1
Cv52023
Dv52023−1
Èa 2-3¼2023·ċ¿ÜŊ·ń§ÂÕÖcŋ
𝑎1;𝑎2;𝑎3;…;𝑎𝑚¼m ~āĂ
ÂýþœŌ𝑎1 ⋅𝑎𝑚
= 𝑡;𝑎2 ⋅𝑎𝑚−1 = 𝑡;…;𝑎𝑚⋅𝑎1 = 𝑡;ö𝑎𝑖⋅𝑎𝑚−𝑖+1 = 𝑡¼t ~]
Â(𝑖= 1,2,⋯,𝑚);ÇÙÅ~“ōÐ|Ņ
”v»Õ;
 8;4;2;
1
2;
1
4;
1
86“ōÐ|Ņ
”vŎt{𝑐𝑛}6 80 * “ōÐ|Ņ
”;𝑡= 2;ÿ𝑐41;
𝑐42;…;𝑐806ŏ~ 2;𝑐80 = 2 |ŏ
;¨
{log2𝑐𝑛} ó n *)~𝑆𝑛;Ç𝑆50 = ¼    Âv
Av210
Bv445
Cv780
Dv1225
 3  
%&'
» 3¼2024·EF·§ČčĎÂŐőŒ6JFœŔŕŖŗ~ĕĖ @¦ŘřŚś;ąŜŝŞŐőŒ şŠ
6šŢFw·ţwŤ;šťŦŧŤï 7 ķcŨũŪsŐőŒ ŝŞ;JFœģŎijæç£ū5 n ¤ŬŒ´
n ąŭŮů
𝑎𝑛;ÿ𝑎1 = 1;𝑎2 = 2;𝑎3 = 5;𝑎4 = 10;𝑎5 = 21;𝑎6 = 42;…;Çū5EŰ 9 ¤ŬŒŮ
ů
ąŭ~¼    Â
Av127
Bv256
Cv341
Dv512
Èa 3-1¼23-24 ń¬·űŲ¿,·ùÂųŴ+ŵŴ6@¦ŴŶų҇ŵÒG¦ˆŷ`Ÿ Źûqź;
6JKŻżŽ Ėž;ſƀƁĘ;9ƒƂmƃƄ;ƅŔĖƆvUÍ@ŶƇƈ~ 1 |Ƈ€ďÁď`
Ŵ;ñĵYųĜƉÒ|Ƈ€ďÁď`;Yų 5 ĵ;ƊĜŵƋŵ5Å9ƌŬ;Çƍ ‹ÆĴ¤…ŷ ŬƎ
Ŵ;ìƏŬƎŴ ƐƑ~¼    Â
Av 2−1
8
Bv2−2
8
Cv 2
8
Dv
1
8
Èa 3-2¼2023·Ü݃Ɠ·@§ÂJFšƔƕŤŝŞhƅƖ¬ąœƗ @¤Ƙµ;Õƙ´ƚ;Í 1;
2;3;…;9 ƛƜ3 × 3 µƝ9;Ÿ ñƞ4ñ4ñœYďƟ¬ 
 )ò…|;ꠋ@¤ 3 ÃƘµ.
@ƠH;Íőơ āĂ
 1;2;3;…;𝑛2ƛƜ𝑛× 𝑛¤µƝ;Ÿ ñƞ4ñ4ñœYďƟ¬ 
 )
ò…|;ì¤āµ`ĞĬ n ÃƘµ. ŝ n ÃƘµ 
 )¼öµƝ9 ´c
 )Â~𝑆𝑛;Õ𝑆3 = 45;×
Ø5Ƣ¶ƣƤ 6¼    Â
Av𝑆6 = 666
      
      
Bv7 ÃƘµï 4 ƞï 4  
ƥƍ(~ 25
Cv8 ÃƘµñƞ4ñ4ñœYďƟ¬ 
 )Ĺ~ 260
Dv9 ÃƘµñƞ4ñ4ñœYďƟ¬ 
 )Ĺ~ 396
Èa 3-3¼23-24 ń5·ŃƦ·ÃßàáÂÎƧŌæTƨYÎƧŌƩƞƪƫ;Ƭo6ƧŌƭÐ ÎÐ
𝐴= {𝑎1,𝑎2,𝑎3, ⋅⋅⋅}üƮƯ;ƮƯÐ~𝐴∗=
𝑎2
𝑎1 , 𝑎3
𝑎2 , 𝑎4
𝑎3 ,⋯;X ï𝑛*~
𝑎𝑛+1
𝑎𝑛;ÆÐ(𝐴∗)∗ ´c*ò
6 3;ÿ𝑎5 = 1;𝑎6 = 27;Ç𝑎1 = ¼    Â
Av
1
9
Bv
1
27
Cv
1
81
Dv
1
243
 4  (
)*+,*-./

» 4¼2024·űư¿Ʊ·Á§Â¢.
{𝐴𝑛};ÆYÓÔ m;𝑛∈𝑁*ÿ𝑚≠𝑛;𝐴𝑚+ 𝐴𝑛6{𝐴𝑛} *;Ç
Ù{𝐴𝑛}~“H 
”v¨
{𝑎𝑛} ó n *)~𝑆𝑛.
(1)Æ𝑆𝑛= 𝑛2 +𝑛;IƲƳ
{𝑎𝑛}6ƴ~“H 
”;ĄƢƵƶƷ
(2)¨{𝑎𝑛}Ƹ6|Ä
Ĥ6“H 
”;ÿ𝑎1 = 6;𝑎2 ∈N∗;𝑎2 > 6;rÄ d ´cƍMňƷ
(3)¨{𝑎𝑛}6|Ä
;ÿYÓÔ𝑛∈𝑁*;𝑆𝑛6{𝑎𝑛} *;rž
{𝑎𝑛}6“H 
”v
Èa 4-1¼2024·ƹư·Á§Â
{𝑎𝑛} ó𝑛* ąmňŝ~𝑀𝑛;ö𝑀𝑛= max{𝑎1,𝑎2, ⋅⋅⋅,𝑎𝑛}Ʒó𝑛* ą
Ćňŝ~𝑚𝑛;ö𝑚𝑛= min{𝑎1,𝑎2, ⋅⋅⋅,𝑎𝑛};ƺ𝑝𝑛= 𝑀𝑛−𝑚𝑛;ĄÍ
{𝑝𝑛}Ù~{𝑎𝑛} “ƻÒ
”v
(1)¨
{𝑝𝑛} “ƻÒ
”~{𝑞𝑛};rž
𝑝𝑛= 𝑞𝑛Ʒ
(2)Æ𝑎𝑛= 2𝑛−3𝑛;rÅƻÒ
{𝑝𝑛} ó𝑛*)v
Èa 4-2¼2024·ŃƦƼÜ·§ČčĎÂŎt
{𝑎𝑛}6ãä×å
;Å
ň~
1,1,2,3,5,8,13, 
21,34 ⋅⋅⋅⋅⋅⋅.ì@
(Õ5ƽƾ µ¶./
𝑎1 = 1，𝑎2 = 1，𝑎𝑛+2 = 𝑎𝑛+1 + 𝑎𝑛 (𝑛∈𝑁*).
{𝑏𝑛}Y=ƿ.
 āĂ
𝑘;ÆĀNāĂ
𝑛Ÿ 𝑏𝑘+𝑛= 𝑏𝑘+ 𝑏𝑛Òă;ÇÙ
{𝑏𝑛}~“𝑘Ãƍǀ
”.
      
      
(1)Ŏt
{𝑐𝑛}ýþ𝑐𝑛= 𝑚𝑎𝑛 （𝑛∈𝑁*，𝑚∈𝑅）.ƲƳ6ƴY∀𝑚∈𝑅;LĀNƿ. āĂ
𝑘;Ÿ 
{𝑐𝑛}
~“𝑘Ãƍǀ
”;ĄƢƵƶ.
(2)¨
{𝑑𝑛} ó𝑛*)~𝑆𝑛= 3𝑛−𝑎 (𝑎≥0);
¼iÂÆ
{𝑑𝑛}~“1Ãƍǀ
”;r£ǁ[œŌ ·
𝑎 ňƷ
¼iiÂN¼iÂ' ó°5;Æ
{𝑓𝑛}ýþ𝑓𝑛=
𝑎𝑛
𝑆𝑛;𝑛∈𝑁*;Åó𝑛*)~𝑇𝑛.žƵ
ņ𝑛∈𝑁*ÿ𝑛≥3d;𝑇𝑛<
𝑎2
1 + 𝑎2
2 + 𝑎2
3 + ⋅⋅⋅⋅⋅⋅+ 𝑎2
𝑛−𝑎𝑛𝑎𝑛+1 +1Òă.
Èa 4-3¼2024·ÜÝǂÉ·Á§ÂÆ
{𝑎𝑛} G*Ĺ~ā
;ÿYÓÔ …ǃÁ*𝑎𝑡−1,𝑎𝑡,𝑎𝑡+1;òý
þ𝑎𝑡−1𝑎𝑡+1 ≤𝑎2
𝑡;ÇÙDŽ
~“Y
0Dž
”;ÆYÓÔ …ǃÁ*𝑎𝑡−1,𝑎𝑡,𝑎𝑡+1;òýþ𝑎𝑡−1 + 𝑎𝑡+1 ≤2
𝑎𝑡ÇÙDŽ
~“Dž
”v
(1)Ŏtā*
{𝑐𝑛}6@¤“Dž
”;ÿ𝑎𝑛= e𝑐𝑛;¼Åe~džƊ]
;𝑛∈N*Â;žƵ

{𝑎𝑛}6@¤“Y

0Dž
”;ÿc𝑎1𝑎10 ≤𝑎5𝑎6Ʒ
(2)Æ=𝑥 {
𝑓(𝑥) = 𝑏1 + 𝑏2𝑥+ 𝑏3𝑥2 + 𝑏4𝑥3cÁ¤LJ8;Å𝑏𝑖> 0 𝑖= 1，2，3，4 vžƵ

𝑏1,𝑏2
,𝑏3,𝑏46@¤“Y
0Dž
”
(3)¨ā*
𝑎0,𝑎1，⋯,𝑎𝑛6@¤“Y
0Dž
”;rž
1
𝑛+1
𝑛
𝑖=0𝑎𝑖
1
𝑛−1
𝑛−1
𝑗=1 𝑎𝑗
≥
1
𝑛
𝑛−1
𝑖=0 𝑎𝑖
1
𝑛
𝑛
𝑗=1𝑎𝑗
 5  
./01'
» 5¼2024·ÜÝ·Á§ÂŎt
{𝑎𝑛} ó n *)~𝑆𝑛;Æ
{𝑎𝑛}ýþ
①
{𝑎𝑛}~cŋ
Ʒ
②
{𝑎𝑛}~ƽLj
Ʒ
③∀𝑘≥2;𝑘∈𝑁∗;∃𝑝,𝑞∈𝑁∗;Ÿ 𝑎𝑘= 𝑎𝑝+ 𝑎𝑞Ʒ
      
      
ÇÙ
{𝑎𝑛}ljc“)01”.
(1)Ŏt𝑆𝑛= 𝑛2 +𝑛(1 ≤𝑛≤100);r
{𝑎𝑛} ,*a;ĄƲƳ
{𝑎𝑛}6ƴljc“)01”Ʒ¼ƲƳ6ƴ
ljc“)01”dˆ>ƢƵƶ;€¢£‰NJÂ
(2)Æē*~ 1 
{𝑎𝑛}ljc“)01”.
¼ⅰÂŏl𝑎𝑛+
𝑆𝑛+1
2 mĆ-;ĄƢƵƶƷ
¼ⅱÂÆ
{𝑎𝑛} Nj*~ 36;r𝑆𝑛 ąĆň.
Èa 5-1¼2024·Êĺưnj·ń§ÂŎt
𝐴:𝑎1,𝑎2,⋯,𝑎𝑛;î𝐴ūï𝑖 1*4ï𝑖 2*4…4ï𝑖 𝑘*𝑖 1 < 𝑖 2
< ⋯< 𝑖 𝑘^Ò
𝐵:𝑎𝑖 1,𝑎𝑖 2,⋯,𝑎𝑖 𝑘;𝐵Ù~𝐴 𝑘*évŝ
𝐵 ´c* )~𝑇(𝐵 )vņ𝑘≥2d;Æ𝐵ýþ
YÓÔ𝑠∈{ 1,2,⋯,𝑘−1 };𝑖𝑠+1−𝑖𝑠= 1;ÇÙ𝐵ljc01𝑃v}.
𝐴 ÓÔ@*ò6𝐴  1 *é;ÿljc0
1𝑃v
(1)ņ𝑛= 4d;ŏl𝐴 ljc01𝑃 é¤
+ˆljc01𝑃 é¤
 mĆ;ĄƢƵƶƷ
(2)Ŏt
𝐴:1 ,2 ,3 , ⋯ ,𝑛   (𝑛≥2)v
¼ⅰ¢.āĂ
𝑘≤
𝑛
2;Y𝐴 𝑘*é𝐵;r´c𝑇(𝐵 ) $ûǍĹňƷ
¼ⅱÂÆ𝐴c𝑚¤ˆŷ ljc01𝑃 é𝐵1,𝐵2,⋯,𝐵𝑚;ýþ
∀  1 ≤𝑖< 𝑗≤𝑚;𝐵𝑖+𝐵𝑗òcǎ*;ÿǎ
*^Ò𝐴 ljc01𝑃 é;r𝑚 ąmňv
Èa 5-2¼2024·ÊĺƦnj·ń§ÂŎt𝐴𝑛:𝑎1,𝑎2,⋯,𝑎𝑛(𝑛≥3)~cŋĂ

;Æ𝐴𝑛ýþ
𝑎𝑖+1−𝑎𝑖∈{𝑝,𝑞}
(𝑖= 1,2,⋯,𝑛−1);Å𝑝;𝑞6õ¤¢. ˆŷǏLJĂ
;ÿ𝑎1 = 𝑎𝑛= 0;ÇÙ𝐴𝑛ljc01𝑇.
(1)Æ𝑝= −1;𝑞= 2;×Ø6ƴĀNljc01𝑇 𝐴5ǐÆĀN;Ǒ£@¤ìí 𝐴5ƷƈĀN;ǒƢƵƶƷ
(2)Æ𝑝= −1;𝑞= 2;ÿ𝐴10ljc01𝑇;rž
𝑎1,𝑎2,⋯,𝑎9>cõ*…ŷƷ
(3)Æ𝑝+ 𝑞= 1;rž
ĀNāĂ
𝑘;Ÿ YÓÔljc01𝑇 𝐴𝑘;òc𝑎1,𝑎2,⋯,𝑎𝑘−1ÓÔõ*ň…ŷ.
      
      
Èa 5-3¼23-24 ń5·ǓŨĻǔ·ùÂŝ𝑅¬ ƍǕ{
𝑓(𝑥) Ǖ{
~𝑓′(𝑥);ýþ𝑥𝑛+1 = 𝑥𝑛−
𝑓(𝑥𝑛)
𝑓′(𝑥𝑛)
(𝑛∈𝑁∗) 
{𝑥𝑛}Ù~{
𝑓(𝑥) “23
”.Ŏt
{𝑥𝑛}~{
𝑓(𝑥) = 𝑥2−𝑥 23
;ÿ
{𝑎𝑛}ý
þ𝑎1 = 2,𝑎𝑛= ln
𝑥𝑛
𝑥𝑛−1,𝑥𝑛> 1.
(1)žƵ
{𝑎𝑛}6|ŏ
Ąr𝑎𝑛Ʒ
(2)¨
{𝑎𝑛} ó𝑛*)~𝑆𝑛;ƈ|a(−1)𝑛⋅𝑡𝑆𝑛−14 ≤𝑆2
𝑛YÓÔ 𝑛∈𝑁∗ǖÒă;r t ūňǗǘ.
 6  23
'
» 6¼2024·ŗƦ·§ČčĎÂ./
Óū
{𝑎𝑛}…ǃ õ*;Æìõ*?Ä ŇYň~ 1;ÇÙ
{𝑎𝑛}ljc“01 1”.Ŏt*
~𝑛 
{𝑎𝑛} ´c* )~𝑀𝑛;ÿ
{𝑎𝑛}ljc“01 1”.
(1)Æ𝑛= 4;ÿ𝑎1 = 0,𝑎4 = −1;Ǒ£´cƍM 𝑀𝑛 ňƷ
(2)Æ𝑎1 = 2024,𝑛= 2023;žƵ
“𝑎2023 = 2”6“𝑎𝑘> 𝑎𝑘+1(𝑘= 1,2,⋯,2022)” ÀoœŌƷ
(3)Æ𝑎1 = 0,𝑛≥2,𝑀𝑛= 0;žƵ
𝑛= 4𝑚‡𝑛= 4𝑚+ 1(𝑚∈𝑁*).
Èa 6-1¼23-24 ń5·½¾·ùÂâ23“ĬƌƟ” µ¶r{
LJ8d;¢£Ǚ“23
”;XNǚǛǚǜ†Ǐ]ŗǝvÅ./6
Y={
𝑓(𝑥);Æýþ(𝑥𝑛+1−𝑥𝑛)𝑓′(𝑥𝑛) + 𝑓(𝑥𝑛) = 0;ÇÙ

{𝑥𝑛}~23
vŎt𝑓(𝑥) = 𝑥4;Õƙ;NǞǟĊ~𝑥1 = 1 8ǠĬ𝑓(𝑥) ƌƟ;ƌƟ+ x Qǡ8 Ǟ
ǟĊ~𝑥2;𝑥2ŔǢ𝑥1üǣ¬Ĉyƭ ‹𝑥3;@€5Ǥ; ‹
{𝑥𝑛}v 
      
      
(1)r
{𝑥𝑛} ,*aƷ
(2)Æ
{𝑛𝑥𝑛} ó n *)~𝑆𝑛;ÿYÓÔ 𝑛∈𝑁∗;ýþ𝑆𝑛≥16−𝜆( 5
6 )
𝑛
;rĂ
  ąĆňƷ¼ǥ
®
0.94 = 0.6561,0.95 ≈0.5905,0.96 ≈0.5314,0.97 ≈0.4783Â
(3)N¼2Â ó°5;¨𝑔(𝑥) =
11
2𝜆𝑓′(𝑥);€Ɵ𝑦= 𝑎𝑥+ 𝑏(𝑏> 0)+ǦƟ𝑦= 𝑔(𝑥)cÿǧcõ¤ǎ8
𝐴(𝑐,𝑑),(ℎ,𝑓);Å𝑐< ℎ;r𝑓(
ℎ
𝑐) ňv
Èa 6-2¼2024·ŗƦǨ·ń§Âŝ𝑅¬ ƍǕ{
𝑓(𝑥) Ǖ{
~𝑓′(𝑥);ýþ𝑥𝑛+1 = 𝑥𝑛−
𝑓(𝑥𝑛)
𝑓′(𝑥𝑛)(𝑛∈𝑁∗)
 
{𝑥𝑛}Ù~{
𝑓(𝑥) “23
”.Ŏt
{𝑥𝑛}~{
𝑓(𝑥) = 𝑥2−𝑥 23
;ÿ
{𝑎𝑛}ýþ𝑎1 = 2,
𝑎𝑛= ln
𝑥𝑛
𝑥𝑛−1,𝑥𝑛> 1.
(1)r𝑎2Ʒ
(2)žƵ
{𝑎𝑛}6|ŏ
Ąr𝑎𝑛Ʒ
(3)¨
{𝑎𝑛} ó𝑛*)~𝑆𝑛;ƈ|a(−1)𝑛⋅𝑡𝑆𝑛−14 ≤𝑆2
𝑛YÓÔ 𝑛∈𝑁∗ǖÒă;r t ūňǗǘ.
Èa 6-3¼23-24 ń¬·ǩűǪǫ·ùNjÂâ23“ĬƌƟ” µ¶r{
LJ8d;¢£Ǚ“23

”;XNǚǛǚǜ†Ǐ]ŗǝvÅ./6
Y={
𝑓(𝑥);Æýþ(𝑥𝑛+1−𝑥𝑛)𝑓′(𝑥𝑛) +𝑓(𝑥𝑛) = 0;Ç
Ù
{𝑥𝑛}~23
vŎt𝑓(𝑥) = 𝑥4;Õƙ;NǞǟĊ~𝑥1 = 1 8ǠĬ𝑓(𝑥) ƌƟ;ƌƟ+ x Qǡ8 
ǞǟĊ~𝑥2;𝑥2ŔǢ𝑥1üǣ¬Ĉyƭ ‹𝑥3;@€5Ǥ; ‹
{𝑥𝑛}v
      
      
  
(1)r
{𝑥𝑛} ,*aƷ
(2)Æ
{𝑛⋅𝑥𝑛} ó n *)~𝑆𝑛;ÿYÓÔ 𝑛∈N ∗;ýþ𝑆𝑛≥16−𝜆5
6
𝑛
;rĂ
𝜆 ąĆňv¼ǥ
®
0.94 = 0.6561;0.95 ≈0.5905;0.96 ≈0.5314;0.97 ≈0.4783Â
 7  
 ./'
» 7¼2024·űưŐű·Á§ÂŎt
{𝑎𝑛}ǎc𝑚(𝑚≥2)*;ÿ𝑎𝑛∈𝑍;Æýþ|𝑎𝑛+1−𝑎𝑛| ≤1
(1 ≤𝑛≤𝑚−1);ÇÙ{𝑎𝑛}~“¥Ǭ
”.ŝ“¥Ǭ
”{𝑎𝑛} ´c* )~𝑆𝑚.
(1)ņ𝑚= 5d;Ǒ£´cýþ𝑎1 = 𝑎5 = 1,𝑆5 = 6 “¥Ǭ
”Ʒ
(2)ņ𝑚= 2000,𝑎1 = 25d;¨𝑝:𝑎2000 = 2024;𝑞:“¥Ǭ
”{𝑎𝑛}~|Ä
.ǒƲƳ𝑝6𝑞 ǭ؜Ō;ĄƢƵ
ƶƷ
(3)ņ𝑎1 = 1,𝑎2𝑘= 0 1 ≤𝑘≤𝑚
2 ,𝑘∈𝑁+ d;r|𝑆𝑚| ąmň.
Èa 7-1¼2024·üļ·§ČčĎÂY=
{𝑎𝑛};./Δ𝑎𝑛= 𝑎𝑛+1−𝑎𝑛(𝑛∈𝑁∗);ýþ𝑎1 = 𝑎2 = 1,Δ(Δ𝑎𝑛)
= 𝑚(𝑚∈𝑅);ŝ𝑓(𝑚,𝑛) = 𝑎1𝑚+ 𝑎2𝑚2 +⋯+ 𝑎𝑛𝑚𝑛;Ù𝑓(𝑚,𝑛)~ƶ
{𝑎𝑛}ƻÒ “𝑚−{
”v
(1)IǑ£“2−{
” 𝑓(2,𝑛);Ąr𝑓(2,3) ňƷ
(2)Æ“1−{
” 𝑓(1,𝑛) ≤15;r n ąmňƷ
(3)ŝ{
𝑆(𝑥) = 𝑥+ 2𝑥2 +⋯+ 𝑛𝑥𝑛;ÅǕ{
~𝑆′(𝑥);žƵ
“𝑚−{
” 𝑓(𝑚,𝑛) = 𝑚2
2 𝑆′(𝑚)−
3𝑚
2
      
      
𝑆(𝑚) + (𝑚+ 1)
𝑛
𝑖=1𝑚′v
Èa 7-2¼2024·űư·§ČčĎÂJFǮŔ
âǯƅǰNDZ š½ǮŪDzŤ@ƕYÃ|Ä
r
)cƃƄ æç;ö“dzŅû”vY=
𝑎1,𝑎2, ⋅⋅⋅,𝑎𝑛, ⋅⋅⋅;①;îïń*ð;ñ@*+Xóô…ǃ@* Ä
^Ò
𝑎11,𝑎12, ⋅⋅⋅,𝑎1(𝑛−1), ⋅⋅⋅;②;ÙDŽ
②~
① @Ãď
;Å𝑎1𝑖= 𝑎𝑖+1−𝑎𝑖
(𝑖= 1,2, ⋅⋅⋅,𝑛−1, ⋅⋅⋅)ƷY=
②;îïń*ð;ñ@*+Xóô…ǃ@* Ä^Ò
𝑎21,𝑎22, ⋅⋅⋅,𝑎2(𝑛−2)
, ⋅⋅⋅;③;ÙDŽ
③~
① ńÃď
;Å𝑎2𝑖= 𝑎1(𝑖+1)−𝑎1𝑖(𝑖= 1,2, ⋅⋅⋅,𝑛−2, ⋅⋅⋅) ⋅⋅⋅⋅⋅⋅”–¬Ĉ
˜¶;ï𝑟ĵ ‹
𝑎𝑟1,𝑎𝑟2, ⋅⋅⋅,𝑎𝑟(𝑛−𝑟), ⋅⋅⋅;④;ÇÙ
④~
① 𝑟Ãď
;Å𝑎𝑟𝑖= 𝑎(𝑟−1)(𝑖+1)
−𝑎(𝑟−1)𝑖(𝑖= 1,2, ⋅⋅⋅,𝑛−𝑟, ⋅⋅⋅);Æ
{𝑎𝑛} 𝑟(𝑟≥2)Ãď
6ǏLJ]
;ÇÙ
{𝑎𝑛}~𝑟Ã|Ä

¼‡Ã|Ä
Âv
(1)ÆÃ|Ä
{𝑎𝑛}~3,4,9,18,31,48, ⋅⋅⋅;r
{𝑎𝑛} ,*aƷ
(2)Æ𝑟Ã|Ä
{𝑏𝑛} ,*a𝑏𝑛= (2𝑛−1)4v
¼ⅰÂr𝑟 ňƷ
¼ⅱÂr
{𝑏𝑛} ó𝑛*)𝑆𝑛v
Ǵ
12 + 22 + ⋅⋅⋅+ 𝑛2 =
𝑛(𝑛+1)(2𝑛+1)
6
v
Èa 7-3¼2024·ǵǶ¿Ǎ·ń§ÂÆ
{𝑐𝑛}ǎc𝑚(𝑚∈𝑁*,𝑚≥3)*;YÓÔ𝑖(𝑖∈𝑁*,𝑖≤𝑚)òc𝑐𝑖𝑐𝑚+1−𝑖
= 𝑆¼𝑆~]
;ÿ𝑆> 0Â;ÇÙ
{𝑐𝑛}6𝑆=𝑚 @¤ŅYÙ
.Ŏt
{𝑎𝑛}6𝑆=𝑚 @¤ŅY
Ù
.
(1)Æ𝑚= 3;𝑎1 = 1;𝑎2 = 2;r𝑎3 ňƷ
(2)Ŏt
{𝑏𝑛}6Ä~𝑑(𝑑≠0) |Ä
;𝑏1 = −11;Æ𝑚= 10;𝑎𝑛=
𝑏𝑛+2
𝑏𝑛;r𝑑)𝑆 ňƷ
      
      
(3)Æ
{𝑎𝑛}6G*Ĺ~āĂ
 ǷǸƽLj
;rž
𝑎𝑚
𝑎1 +
𝑎𝑚−1
𝑎2 + ⋅⋅⋅+
𝑎2
𝑎𝑚−1 +
𝑎1
𝑎𝑚<
5𝑆
3 .
@4Ƿ
1v¼2024·½¾ǹŊ·§ČčĎÂŎt
{𝑎𝑛} G*Ĺ~ā
;𝑎1 = 1,𝑎𝑛+1−𝑎𝑛=
1
𝑎𝑛+1+𝑎𝑛;Æ[𝑥]Ǻƚˆǻy
𝑥 ąmĂ
;Ç[𝑎1] + [𝑎2] +⋯+ [𝑎100] = ¼   Â
Av615
Bv620
Cv625
Dv630
2v¼2024·¬Ł·§ČčĎÂŎt
{𝑎𝑛}ˆ6]
;ó𝑛*)~𝑆𝑛;ÿ𝑎1 > 0vÆYÓÔāĂ
𝑛;ĀNā
Ă
𝑚;Ÿ |𝑎𝑛−𝑆𝑚| ≤𝑎1;ÇÙ{𝑎𝑛}6“ƍǼ
”vU¢£õ¤ǽ
①ĀN|Ä
{𝑎𝑛}6“ƍǼ
”Ʒ
②ĀN|ŏ
{𝑎𝑛}6“ƍǼ
”vÇ5ƲƳāƿ 6¼    Â
Av①+②Ĺ~÷ǽ
Bv①+②Ĺ~Ǿǽ
Cv①~÷ǽ;②~Ǿǽ
Dv①~Ǿǽ;②~÷ǽ
3v¼2024·EF·§ČčĎÂÍāĂ
 n s~õ¤āĂ
𝑘1;𝑘2 Ņ;ö𝑛= 𝑘1𝑘2;ņ𝑘1;𝑘2õ
Ä ŇY
ňąĆd;JKÙÅ~ąʼns.Õ12 = 1 × 12 = 2 × 6 = 3 × 4;Å3 × 4ö~ 12 ąʼns;ņ𝑘1;𝑘2
6 n ąʼnsd;./𝑓(𝑛) = |𝑘1−𝑘2|;Ç
{𝑓(2𝑛)} ó 2024 * )~¼    Â
Av21011−1
Bv21011
Cv21012−1
Dv21012
4v¼2024·ÜÝÜļ·Á§ÂÆ*
Ĺ~𝑛(𝑛≥2,𝑛∈𝑁*) õ¤
{𝑎𝑛},{𝑏𝑛}ýþ𝑎𝑘−𝑏𝑘= 𝑘(𝑘= 1,2,⋯,𝑛);ÿ
i[{𝑎1,𝑎2,⋯,𝑎𝑛,𝑏1,𝑏2,⋯,𝑏𝑛} = {1,2,3,⋯,2𝑛};ÇÙ
{𝑎𝑛},{𝑏𝑛}6@Y“𝑛*ǿȀ
”v¨
{𝑎𝑛},{𝑏𝑛}6@
Y“4 *ǿȀ
”;Çìí “4 *ǿȀ
”c¼    ÂYv
Av5
Bv6
Cv7
Dv8
5v¼2024·½¾¿À·Á§ÂY=
{𝑎𝑛};}.Δ𝑎𝑛~
{𝑎𝑛} @Ãď;ÅΔ𝑎𝑛= 𝑎𝑛+1−𝑎𝑛(𝑛∈N*);
}.Δ𝑘𝑎𝑛~
{𝑎𝑛} k Ãď;ÅΔ𝑘𝑎𝑛= Δ𝑘−1𝑎𝑛+1−Δ𝑘−1𝑎𝑛(𝑛∈N*)vÆ𝑎𝑛=
𝑛(𝑛−1)(2𝑛−1)
6
;ÇΔ2𝑎6 =
¼    Â
Av7
Bv9
Cv11
Dv13
6v¼2024·¬ŁłŃ·ń§Â
{𝑎𝑛};𝑆𝑛6Åó𝑛* );ÆYÓÔāĂ
𝑛;LĀNāĂ
𝑚;Ÿ 𝑆𝑛=
      
      
𝑎𝑚;ÇÙ
{𝑎𝑛}~“Î
”.UcÕ5õ¤ǽ
①|ŏ
{2𝑛}~“Î
”Ʒ②YÓÔ |Ä
{𝑎𝑛};
LĀNõ¤“Î
”{𝑏𝑛}){𝑐𝑛};Ÿ 𝑎𝑛= 𝑏𝑛+ 𝑐𝑛.Ç5*āƿ 6¼    Â
Av①~÷ǽ;②~÷ǽ
Bv①~÷ǽ;②~Ǿǽ
Cv①~Ǿǽ;②~÷ǽ
Dv①~Ǿǽ;②~Ǿǽ
7v¼23-24 Á5·üļ·ÃßàáÂ./
ýþ 
𝑎𝑛+2
𝑎𝑛+1:
𝑎𝑛+1
𝑎𝑛= 𝑞(𝑞  ~]
;𝑛∈N*Â 
{𝑎𝑛} Ù~ńÃ|ŏ

;𝑞~ńÃŏ.ŎtńÃ|ŏ
∣𝑎𝑛} ńÃŏ~ 2,𝑎1 = 1,𝑎2 =
2;ǟ 𝑎𝑛> 2024 Òă ąĆ
āĂ
𝑛~¼    Â
Av7
Bv8
Cv9
Dv10
8v¼2024·ÊĺƦnj·ń§Â¨ȁŋā

{𝑎𝑛};ÕÖYÓÔ āĂ
𝑛;òĀNȂ@ āĂ
𝑚;Ÿ 
𝑎𝑚= 𝑎1 + 𝑎2 + 𝑎3 +⋯+ 𝑎𝑛;×ØÙ{𝑎𝑛}~9)
;Ąƺ𝑏𝑛= 𝑚;Ù{𝑏𝑛}~{𝑎𝑛} fg
;Ǽ    Â
AvÆ{𝑎𝑛}~|Ä
;Ç{𝑎𝑛}~9)

BvÆ{𝑎𝑛}~|ŏ
;Ç{𝑎𝑛}~9)

CvÆ9)
{𝑎𝑛}~ƽLj
;ÇÅfg
{𝑏𝑛}~ƽLj

DvÆ9)
{𝑎𝑛} fg
{𝑏𝑛}~ƽLj
;Ç{𝑎𝑛}~ƽLj

ń4ȃ
9v¼2024·ŃƦȄȅ·Á§ÂÆcŋĂ

𝐴𝑛:𝑎1,𝑎2,⋯𝑎𝑛(𝑛≥3)ýþ
𝑎𝑖+1−𝑎𝑖∈{−1,2}(i = 1,2,⋯,𝑛−1);
ÿ𝑎1 = 𝑎𝑛= 0;ÇÙ𝐴𝑛ljc01𝑇.Ǽ     Â
AvĀNljc01𝑇 𝐴4
BvĀNljc01𝑇 𝐴5
CvÆ𝐴10ljc01𝑇;Ç𝑎1,𝑎2,⋯,𝑎9ŕŭcõ*…ŷ
DvĀNāĂ
𝑘;Ÿ YÓÔljc01𝑇 𝐴𝑘;c𝑎1,𝑎2,⋯,𝑎𝑘−1ÓÔõ*ň…ŷ
10v¼2024·üļ·§ČčϨ{𝑎𝑛}6G*~ā ȁŋ
;ÆY=∀𝑛∈𝑁*;𝑎2
𝑛+1−𝑎2
𝑛= 𝑑¼d
~ǏLJ]

Â;ÇÙ
{𝑎𝑛}~|µÄ
v×ؼ   Â
AvÆ{𝑎𝑛}6|µÄ
;Ç 𝑎2
n 6|Ä

Bv
{2n}~|µÄ

CvÆ{𝑎𝑛}6|µÄ
;Ç
{𝑎𝑛+1−𝑎𝑛}ĀNĆ= 1 *
DvÆ{𝑎𝑛}6|µÄ
;ÇĀNāĂ
 n;Ÿ 
𝑛
𝑖=1
1
𝑎𝑖> 2024
11v¼2024·ǩűȆä·§ČčĎÂŎt
{𝑢𝑛};Åó n *)~𝑆𝑛;ÆĀN]
𝑀> 0;YÓÔ 𝑛∈N∗;ǖ
      
      
c|𝑢𝑛+1−𝑢𝑛| + |𝑢𝑛−𝑢𝑛−1| +⋯⋯+ |𝑢2−𝑢1| ≤𝑀;ÇÙ{𝑢𝑛}~𝐵−
.Ç5Ƣ¶āƿ 6¼    Â
AvÆ{𝑢𝑛}6( 1 ~ē*;𝑞(|𝑞|＜1)~ŏ |ŏ
;Ç{𝑢𝑛}~𝐵−

BvÆ{𝑢𝑛}~𝐵−
;Ç{𝑆𝑛}ȇ~𝐵−

CvÆ{𝑆𝑛}~𝐵−
;Ç{𝑢𝑛}ȇ~𝐵−

DvÆ{𝑎𝑛},{𝑏𝑛}Ĺ~𝐵−
;Ç{𝑎𝑛⋅𝑏𝑛}ȇ~𝐵−

Á4ƛǛ
12v¼2024·űŲȈƓ·§ČčĎÂY=cŋ
{𝑎𝑛};î
{𝑎𝑛}ūï𝑖1*､ï𝑖2*､⋯､ï𝑖𝑚*
(𝑖1 < 𝑖2 < ⋯< 𝑖𝑚);ÏĵÑ^Ò
{𝑏𝑘};Å𝑏𝑘= 𝑎𝑖𝑘,1 ≤𝑘≤𝑚;ÇÙ
{𝑏𝑘}~{𝑎𝑛} @¤é;Ù{𝑏𝑘}
G*?)~{𝑎𝑛} @¤é)v}.

{𝑎𝑛} ÓÔ@*ò6{𝑎𝑛} évÇ
1,2,4,8,16,32 ´cé
) )~      
v
13v¼2023·ȉȊűȋȌȍ·§ČčĎÂ23Ŕ¶ĤÙ23——ȎĠȏµ¶;X623N 17 ƅȐ°£ @
¦N·
i¬BȑrsµƭȒ @¦µ¶.ljȓŮůÕ5ƙƚ
¨ r 6{
𝑦= 𝑓(𝑥) @¤LJ8;ÓÔū𝑥0
Ĭ~ r ȔȕBȑň;N8(𝑥0,𝑓(𝑥0))ĬǦƟ𝑦= 𝑓(𝑥) ƌƟ𝑙1;¨+𝑙1Q x ǡ8 ǞǟĊ~𝑥1;ĄÙ𝑥1~ r
 1 ĵBȑňƷN8(𝑥1,𝑓(𝑥1))ĬǦƟ𝑦= 𝑓(𝑥) ƌƟ𝑙2;¨+𝑙2Q x ǡ8 ǞǟĊ~𝑥2;Ù𝑥2~ r 2 ĵB
ȑň.@ƠH;N8(𝑥𝑛,𝑓(𝑥𝑛))(𝑛∈N)ĬǦƟ𝑦= 𝑓(𝑥) ƌƟ𝑙𝑛+1;ŝ𝑙𝑛+1+ x Qǡ8 ǞǟĊ~𝑥𝑛+1;ĄÙ𝑥𝑛+1
~ r 𝑛+ 1ĵBȑň.¨𝑓(𝑥) = 𝑥3 +𝑥−3(𝑥≥0) LJ8~ r;ū𝑥0 = 0;Ç r 1 ĵBȑň~      
ƷÆ𝑥𝑛~
r n ĵBȑň;¨𝑎𝑛=
3𝑥3
𝑛+𝑥𝑛
2𝑥3
𝑛+3 ;𝑛∈N∗;
{𝑎𝑛} ó n *Ņ~𝑇𝑛.ÆÓÔ𝑛∈N∗;𝑇𝑛> 𝜆ǖÒă;ÇĂ
𝜆
 ąmň~      
.
14v¼2024·Êĺ,Ɠ·Á§ÂÆ
{𝑏𝑛}4{𝑐𝑛}Ĺ~ȖƝLj
;ÿYÓÔāĂ
 n;òĀNāĂ
 m;Ÿ 
𝑏𝑚∈[𝑐𝑛,𝑐𝑛+1];ÇÙ
{𝑏𝑛}~
{𝑐𝑛} “M 
”vŎt
{𝑎𝑛} ó n *)~𝑆𝑛;Ç5‰NJāƿ 
6      
v
①ĀN|Ä
{𝑎𝑛};Ÿ {𝑎𝑛}6{𝑆𝑛} “M 
”
②ĀN|ŏ
{𝑎𝑛};Ÿ {𝑎𝑛}6{𝑆𝑛} “M 
”
③ĀN|Ä
{𝑎𝑛};Ÿ {𝑆𝑛}6{𝑎𝑛} “M 
”
      
      
④ĀN|ŏ
{𝑎𝑛};Ÿ {𝑆𝑛}6{𝑎𝑛} “M 
”
½4sȗ
15v¼2024·ǩű·§ČčĎÂ./:[𝑥]Ǻƚ𝑥 Ă
ۏ;{𝑥}Ǻƚ𝑥 Ć
ۏ;»Õ[1.2] = 1,{1.75} = 0.75.

𝑎𝑛ýþ𝑎𝑛+1 =
[𝑎𝑛]
{𝑎𝑛}
(𝑎𝑛∉Z)
𝑎𝑛
(𝑎𝑛∈Z)
 
Å𝑎1 = 𝑚.ÆĀN𝑘∈N+;Ÿ ņ𝑛> 𝑘d;𝑎𝑛= 𝑎𝑛+1ǖÒă;ÇÙ
𝑚
~Ș„
.
(1)șǑ£ņ𝑚=
2,𝑚=
5
3d𝑎1,𝑎2,𝑎3,𝑎4 ň.
(2)žƵ: 𝑡2 + 1(𝑡∈N+)6Ș„
(3)Æ𝑚~m= 1 c
.ÿ𝑚∉Z.rž:𝑚~Ș„
16v¼23-24 ń5·ÜÝȚÜ·ùNjÂÕÖȁŋ
{𝑎𝑛}ýþ“YÓÔāĂ
𝑖,𝑗(𝑖≠𝑗);òĀNāĂ
𝑘;Ÿ 
𝑎𝑘= 𝑎𝑖⋅𝑎𝑗”;ÇÙ
{𝑎𝑛}ljc“01𝑃”.
(1)Æ|ŏ
{𝑎𝑛} ó𝑛*)~𝑆𝑛;ÿŏ𝑞> 1,𝑆2 = 12,𝑆4 = 120;rž

{𝑎𝑛}ljc“01𝑃”Ʒ
(2)Æ|Ä
{𝑏𝑛} ē*𝑏1 = 1;Ä𝑑∈𝑍;rž

{𝑏𝑛}ljc“01𝑃”;ņÿțņ𝑑∈𝑁Ʒ
(3)ÕÖG*Ĺ~āĂ
 ȁŋ|ŏ
{𝑐𝑛}ljc“01𝑃”;ÿ213,512,415,1012½¤
Ûcõ¤£UN

{𝑐𝑛};r𝑐1 ´cƍMūň?).
17v¼2024·ȉȊű·Á§ÂÕÖ n *cŋ
{𝑎𝑛}ýþ𝑎1 = 𝑎𝑛;𝑎2 = 𝑎𝑛−1;…;𝑎𝑛= 𝑎1;ö𝑎𝑖= 𝑎𝑛−𝑖+1
(𝑖= 1,2,⋯,𝑛);ÇÙcŋ
{𝑎𝑛}~“YÙ
”.
(1)¨
{𝑏𝑛}6*
~ 7 “YÙ
”;Å𝑏1,𝑏2,𝑏3,𝑏4Ò|Ä
;ÿ𝑏2 = 3,𝑏5 = 5;­ĵǑ£
{𝑏𝑛} 
ñ@*Ʒ
(2)¨
{𝑐𝑛}6*
~2𝑘−1(𝑘∈𝑁∗ÿ𝑘≥2) “YÙ
”;ÿýþ|𝑐𝑛+1−𝑐𝑛| = 2;ŝ𝑆𝑛~
{𝑐𝑛} ó𝑛*
).
      
      
①Æ𝑐1;𝑐2;…;𝑐𝑘^ÒǷǸƽLj
;ÿ𝑐𝑘= 2023.ņ𝑘~Ȝňd;𝑆2𝑘−1ū ąmň?
②Æ𝑐1 = 2024;ÿ𝑆2𝑘−1 = 2024;r𝑘 ąĆň.
18v¼2024·ǵǶȝƓ·§ČčĎÂÆȁŋ
{𝑎𝑛}ýþ
Y=∀𝑛∈N*，𝑎2
𝑛+1−𝑎2
𝑛= 𝑝;Å𝑝~]
;ÇÙ
{𝑎𝑛}~𝑃
v
(1)Æ@¤ŏ~𝑞 |ŏ
{𝑥𝑛}~“𝑃
”;r𝑞 ňƷ
(2)Æ𝑎1 = 1，𝑝= 2，{𝑦𝑛}6ē*~ 1;ŏ~ 3 |ŏ
;N𝑦𝑘+𝑦𝑘+1?ŀ­ĵȞƜ
 𝑎2
𝑛 𝑘*^
Ò
{𝑐𝑛}:𝑦1,𝑎2
1,𝑦2,𝑎2
2,𝑎2
3,𝑦3,𝑎2
4,𝑎2
5,𝑎2
6,𝑦4, ⋅⋅⋅⋅⋅⋅;r
{𝑐𝑛}ó 30 * )𝑆30v
(3)Æ@¤“𝑃
"{𝑎𝑛}ýþ𝑎1 = 2,𝑎2 = 2 2,𝑎𝑛> 0;¨

1
𝑎𝑛 ó𝑛*)~𝑇𝑛v6ƴĀNāĂ
𝑚,𝑘;Ÿˆ
|a𝑇𝑛>
𝑚𝑛+ 𝑘−1Y@ƌ𝑛∈𝑁∗òÒăǐÆĀN;r£𝑚,𝑘 ňƷƈĀN;ƢƵƶv
19v¼2024·ǵǶȟȠ·§ČčĎÂY=
{𝑎𝑛};
{𝑎𝑛+1−𝑎𝑛}Ù~
{𝑎𝑛} Ä
‡@ÃÄ
.{𝑎𝑛}Ä

 Ä
;Ù~{𝑎𝑛} ńÃÄ
.@ƠH;{𝑎𝑛} 𝑘ÃÄ
 Ä
;Ù~{𝑎𝑛} 𝑘+ 1ÃÄ
.ÕÖ
{𝑎𝑛} 𝑘ÃÄ
~]
;ĩ𝑘−1ÃÄ
ˆ6]
;×Ø{𝑎𝑛}ĪÙ~𝑘Ã|Ä
.
(1)Ŏt 20;24;26;25;20 6@¤𝑘Ã|Ä
{𝑎𝑛} ó 5 *.r𝑘 ňȡ𝑎6Ʒ
(2)žƵ
ńÃ|Ä
{𝑏𝑛} ,*a~𝑏𝑛= 𝑏1 + (𝑛−1)(𝑏2−𝑏1) +
1
2(𝑛−1)(𝑛−2)(𝑏3−2𝑏2 + 𝑏1)Ʒ
(3)žƵ
Æ
{𝑐𝑛}6𝑘Ã|Ä
;Ç{𝑐𝑛} ,*a6𝑛 𝑘ĵȃ*a;ö𝑐𝑛=
𝑘
𝑖=0𝜆𝑖𝑛𝑖¼Å𝜆𝑖¼𝑖= 0，
1，⋯，𝑘Â~]·
Â