专题6.2等差数列及其前n项和十一大题型（原卷版）_11.4-2_2026-11-04

      
      
专题6.2 等差数列及其前n 项和【十一大题型】

 1  
 !"#$%& ........................................................................................................................3
 2  
 !'()*+ ........................................................................................................................4
 3  
 !,-./ ........................................................................................................................4
 4  
 !012 ............................................................................................................................5
 5  
 3 n 14!,- ....................................................................................................................6
 6  
 !3 n 14!56 ................................................................................................................6
 7  
 !78/ ............................................................................................................................7
 8  
 !9:1;<= ................................................................................................................8
 9  >?@6!
 = ....................................................................................................................9
 10  
 A!BCD= ................................................................................................................10
 11  )
 E!(FGHI= ........................................................................................11
1G
 .J3 n 14
KLM
NOP
HQR
(1)S
 !TU
4012!VF
(2)WXYZ[
 
!3 n 142\S

 !012)3 n
142!]
(3)^_`a=HbA\
cd
 !]\Y
Sef/!=
(4)ag
 )hi
j
!]
2022 klmnop
q
r 13 \5 Q
2023 k I o
r 7 \
5 Q
2023 kⅡo
r 18 \
12 Q
2024 k I o
r 19 \
17 Q
2024 kⅡo
r 12 \
5 Q

 s!tKuv\wx
!yuvzh.{|}k!
H~€\
 !"#$P&4"
#,-G
 !A1,-G'(s
!tK\‚Lƒ„G…†
!‡2\ˆ‰Š‹Œ
 !
*+GM4.Ž/s!
K\h‘d_S’A\ˆ‰A.
“k”•A‘d
 !(
FGHI\ˆ‰Š–\—L˜™M
S. 
š›K 1  
 !TU
1œ
 !TU
h\žŸh 
 {r 2 1¡\¢h1)£!3h1!¤x¥h y
\¦§¨ 
 ©ª
«
 \¨ y
ª«
 !\y¬­ d ®¯.
2œA1
°± 
 a,A,b ²C!
 ³ƒ€Cs578!
 \¨´ A ª« a ) b !A1\µE
      
      
2A=a+b.¶z\· 2A=a+b\µ a,A,b ± 
C
 .
3œ
 !012

 !012¸
=
+(n-1)d\JA
¸¹1\d ¸.
4œ
 !8º,
°
 !0124h»j
!]³š
 !8º,¼ d ½¾.
¿À d>0 ´\
 ¸ÁÂ
 \žÃ¿Ä¯Œ
ÅÀ d<0 ´\
 ¸ÁÆ
 \žÃÅįŒ
ÇÀ d=0 ´\
 ¸y
 \žÃÇį.
ÈÉ\Ê<¸Ë6\
 ¤ÌgsÍÎ
 .
5œ
 !,-
Ï{
}¸
 \¸ d\µ
(1)· m+n=p+q(m,n,p,q
)\µ
+
=
+
.
(2)
 {
+b}( ,b sy
)s¸ d !
 .
(3)·{
}s¸ d'!
 \{
}){
}!1
hÐ\µ
 {
+
 (
,
¸y
)s¸
d+
d'!
 .
(4)ÑÒC
 Ó¸ m !1
,
,
,
(k,m
)²C¸ md !
 .
(5)_
 {
}A\·
=m\
=n\m≠n\µE
=0.
š›K 2  
 !"#%&!SÔÕ
1œ
 !"#%&!֖MS×Ø
(1)
 !012.3 n 142ÙÚ.Û $ a1\an\d\n\Sn\šJA± ©^MÜÝÖ \
adÞßà!×áSe=.
(2)
 !01243 n 142_SA¡âã$äåæ\ç a1 4 d s
 !Ö "#$\
£è®¯éš4êšsyßë.
š›K 3  
 !'(!ßë)ì<
1œ*+
 s
 !‚Lßë
(1)(Fë
@x ní2 !îVïð
\ñ* an-an-1 ¸¥hy
.òæë\óx an-1 ! an äô an-an-1\
_7õâ(6.
(2)A1ë
ñ* 2an-1=an+an-2pní3\nöN*q¤CD.
2œ'(h 
 s
 ÷yâ!ì<
(1)012
an=pn+qpp,q ¸y
q
s
 .
(2)3 n 142
Sn=An2+BnpA,B ¸y
q
s
 .
      
      
=!5ø'(÷sù(F.
š›K 4  
 .J3 n 14!,-./
1œ1!,-
_
 
A\· m+n=p+q(m,n,p,qöN*)\µ am+an=ap+aq.
2œ4!,-
_
 
A\Sn ¸J3 n 14\µ
(1)S2n=n(a1+a2n)=…=n(an+an+1)Œ
(2)S2n-1=(2n-1)anŒ
(3)û» k 14C
 \ò Sk,S2k-Sk,S3k-S2k,úC
 .
3œM
 3 n 14!56!yßë
(1)ü1ã	ë
ù
 !8º,\M‘JýþÿĀ1\āĂù,-MJýþÿĀ1\ă³M
õ4!56Œ
(2)Ą»j
ë
ù̸ą!
 !3 n 14 Sn=An2+BnpA,B ¸y
\AĆ0q¸Ą»j
\
0楻j
!,-M56.
(3)Ì2²ë
ĈĉÀ Sn 5–´\E
\SÉÌ2²Ċ( n !ċČ\č
çĊ( n !64@/ Sn !6(ò Sn 5–6)\Ďď³M Sn !5Đ6.
ßëđĒ)ēì
1œéš
 {
}!012s
= pn+q(JA p\q ¸y
)\µ
 {
}h(s
 \Ó¸
p.
2œ_
 {
}A\a1>0\ d<0\µ Sn Ĕ_5–6Œ· a1<0\ d>0\µ Sn Ĕ_5Đ6.
3œ
 {
}!8º,
À d>0 ´\{
}sÁÂ
 ŒÀ d<0 ´\{
}sÁÆ
 ŒÀ d=0 ´\
{
}sy
 .
4œ
 {
}s
 
( A, B ¸y
).
 1  
 !"#$%&
ĕ 1p2024·ĖėĘę·ĚěĜĝq·
 {𝑎𝑛}Ğğ𝑎𝑛+ 𝑎𝑛+1 = 4𝑛+ 1\µ𝑎1 = p    q
Aœ3
Bœ
3
2
Cœ1
Dœ
1
2
ã2 1-1p2024·ĠġĢ(·±Ěqéš_
 {𝑎𝑛}A\𝑎1 = 1\𝑑> 0.·
 𝑎2
𝑛−4
𝑛
ģs
 \µ𝑑= p    q
Aœ1
Bœ2
Cœ3
Dœ4
ã2 1-2p2024·uĤĥĦħ·±ĚqÏ𝑆𝑛¸
 {𝑎𝑛}!3 n 14\·𝑆5 = 4𝑎1\𝑎1 > 0\·𝑛> 1´\
𝑆𝑛= 𝑎𝑛\µ𝑛xp    q
      
      
Aœ11
Bœ12
Cœ20
Dœ22
ã2 1-3p2024·ġĨ·ĚěĜĝqĩ
 {𝑎𝑛}!¸𝑑\3𝑛14¸𝑆𝑛\·𝑎5 + 𝑎11 = 62\Ó𝑆13
= 351\µĪ
 !𝑑¸p    q
Aœ3
Bœ4
Cœ5
Dœ6
 2  
 !'()*+
ĕ 2p2024·īġĬĭ·ĚěĜĝqéš
 {𝑎𝑛}\µ“𝑎𝑛−2 + 𝑎𝑛+2 = 2𝑎𝑛𝑛≥3，𝑛∈𝑁∗”s“
 {𝑎𝑛}s

 ”!p    q
AœĮQÌįLİı
BœįLÌĮQİı
CœĮLİı
DœIJÌĮQģÌįLİı
ã2 2-1p2024·ijĴĵĶ·ĚěĜĝqÏý

 {𝑎𝑛}!3𝑛14¸𝑆𝑛\Ó𝑆𝑛=
1
2 𝑎𝑛+ 1
𝑎𝑛(𝑛∈𝑁∗)\µ
p    q
Aœ{𝑎𝑛}s
        
Bœ{𝑆𝑛}s
        
Cœ{𝑎𝑛}8ºÁ       
Dœ{𝑆𝑛}8ºÁÂ
ã2 2-2p2023·ķ·hĚqĸą
 {𝑎𝑛}Ğğ(𝑎𝑛+1−𝑎𝑛)(2𝑎𝑛+1−𝑎𝑛+2) = 𝑎𝑛(𝑎𝑛+2−𝑎𝑛+1)(𝑛∈𝑁*)\Ó
𝑎1 = 1,𝑎2 = 2œ
(1)Ï𝑏𝑛=
𝑎𝑛
𝑎𝑛+1−𝑎𝑛\*+

 {𝑏𝑛}s
 Œ
(2)Ï𝑐𝑛=
1
𝑎𝑛𝑎𝑛+1\M{𝑐𝑛}!3𝑛14𝑇𝑛œ
ã2 2-3p2024·Ĺĺĺij·ĚěĜĝqéš
 {𝑎𝑛}!3𝑛1!Ļĩ¸𝑇𝑛\ÓĞğ
1
𝑇𝑛=
𝑎𝑛−1
2𝑎𝑛.
(1)*+

 {𝑇𝑛}¸
 Œ
(2)Ï𝑏𝑛=
1
𝑇𝑛𝑇𝑛+1\M
 {𝑏𝑛}!3𝑛14𝑆𝑛.
      
      
 3  
 !,-./
ĕ 3p2024·ļĺ%Ľ·±Ěqéš
 {𝑎𝑛}s
 \
1
2𝑎3−𝑎5 = 2\µ𝑎5 + 𝑎10−𝑎8 = p    q
Aœ4
Bœ−2
Cœ−4
Dœ−8
ã2 3-1p2024·lm·ĚěĜĝq_
 {𝑎𝑛}A\éš2𝑎𝑛+1 = 𝑎𝑛+ 𝑎𝑛+2(𝑛∈N∗)\Ó𝑎5 + 𝑎8 = 16\µ
𝑎1 + 𝑎2 +… + 𝑎12 = p    q
Aœ256
Bœ196
Cœ144
Dœ96
ã2 3-2p2024·lm·ĚěĜĝqéš
 {𝑎𝑛}Ğğ𝑎1𝑎3 + 𝑎2𝑎7 + 𝑎3𝑎9 + 𝑎7𝑎8 = 100\µ𝑎5 =
p    q
Aœ
5
2
Bœ5
Cœ5 āľ5
Dœ
5
2ā−
5
2
ã2 3-3p2024·ĿĺŀŁ·ĚěĜĝqéš
 {𝑎𝑛}!̸ 0\𝑎2024 = 0\ł(ýŃ
 m\ńõ
@îV!𝑛∈𝑁*p𝑛< 𝑚Ó𝑚> 2q¤E𝑎1 + 𝑎2 +⋯+ 𝑎𝑛= 𝑎1 + 𝑎2 +⋯+ 𝑎𝑚−𝑛CD\µ m !6¸p    q
Aœ4047
Bœ4046
Cœ2024
Dœ4048
 4  
 !012
ĕ 4p2024·Ņņ·ĚěĜĝqéš𝑆𝑛¸ý1
 {𝑎𝑛}!3𝑛14\𝑎1 = 3Ó𝑆𝑛+ 𝑆𝑛+1 =
1
2𝑎2
𝑛+1−
3
2\µ𝑎𝑛=
      
.
ã2 4-1p23-24 ĄÑ·ĿŇňʼn·ŊŋŌōqéš
 {𝑎𝑛}!3𝑛14𝑆𝑛= 𝑛2 +𝑛+ 𝑐pJA𝑐¸y
\
𝑐∈𝑅q\Ŏ‘ń{𝑎𝑛}¸
 !h 012𝑎𝑛=      
.
ã2 4-2p2024 ±·ĿŇ·Ōōqéš
 {𝑎𝑛}¸̸ą!
 \𝑆7 = 77\ÓĞğ𝑎112 =
𝑎1 ⋅𝑎61\
p1qM
 {𝑎𝑛}!012Œ
p2q·
 {𝑏𝑛}Ğğ
1
𝑏𝑛+1−
1
𝑏𝑛= 𝑎𝑛(𝑛∈𝑁∗)\Ó𝑏1 =
1
3\M
 {𝑏𝑛}!3 n 14𝑇𝑛.
ã2 4-3p2024 ±·lm·Ōōqéš
 {𝑎𝑛},{𝑏𝑛}\JA
 {𝑎𝑛}s
 \ÓĞğ𝑏𝑛−𝑎𝑛= (−1)𝑛
𝑛2\𝑎1 + 𝑏1 = 1\𝑎2 + 𝑏2 = 8,𝑛∈𝑁*œ
(1)M
 {𝑎𝑛}4{𝑏𝑛}!012Œ
      
      
(2)·𝑐𝑛=
1
𝑎𝑛𝑎𝑛+1\M
 {𝑐𝑛}!3𝑛14𝑆𝑛Œ
 5  
 3 n 14!,-
ĕ 5p2024·ĹĺŏĶ·ĄĚqéš
 {𝑎𝑛}!3𝑛14¸𝑆𝑛\·𝑆4 = 2\𝑆8 = 12\µ𝑆20 = p    q
Aœ30
Bœ58
Cœ60
Dœ90
ã2 5-1p2024·lm·ĚěĜĝqéš
 {𝑎𝑛},{𝑏𝑛}!3𝑛14QŐ¸𝑆𝑛,𝑇𝑛\Ó
𝑆𝑛
𝑇𝑛=
𝑛−1
𝑛+1\µ
𝑎6
𝑏8 =
p    q
Aœ
5
16
Bœ
7
16
Cœ
11
16
Dœ
5
8
ã2 5-2p2024·Ņņőļ·hĚqÏ
 {𝑎𝑛}!3𝑛14𝑆𝑛\·𝑆3 = 9\𝑆6 = 36\µ𝑎7 + 𝑎8 + 𝑎9 =
p    q
Aœ18
Bœ27
Cœ45
Dœ63
ã2 5-3p2024·ĿŇŒļ·ĚěĜĝqÏ
 {𝑎𝑛}\{𝑏𝑛}!3𝑛14QŐ¸𝑆𝑛\𝑇𝑛\·@îVýŃ
𝑛
¤E
𝑆𝑛
𝑇𝑛=
2𝑛−3
4𝑛−3\µ
𝑎3
𝑏4+𝑏8 +
𝑎9
𝑏5+𝑏7 = p    q
Aœ
3
7
Bœ
5
21
Cœ
19
41
Dœ
19
40
EœœÌs
 6  
 !3 n 14!56
ĕ 6p2024·ŔŕŖŗŘ·ĄĚq
 {𝑎𝑛}A\𝑎1 > 0\𝑆7 = 𝑆9\µńõ3 n 1!45–! n 6¸
p    q
Aœ7
Bœ8
Cœ9
Dœ10
ã2 6-1p2024·řŚĖśŜŝ·ĚěĜĝqéš{𝑎𝑛}s
 \𝑆𝑛sJ3𝑛1!4\µÑ ì<Şş!
sp    q
Aœ·𝑎𝑛= 2𝑛−25\µ𝑆𝑛Š5Đ6´𝑛!6¸ 12
Bœ·𝑎𝑛= −3𝑛+ 27\µ𝑆𝑛!5–6¸ 108
Cœ·𝑆13 = 𝑆17\µįE𝑆30 = 0
Dœ·¹1𝑎1 > 0\𝑆6 = 𝑆12\µ𝑆𝑛Š5Đ6´𝑛!6¸ 9
      
      
ã2 6-2p2024·lm·ĚěĜĝqĩ
 {𝑎𝑛}!3𝑛14¸𝑆𝑛\¸𝑑\éš𝑎4 = −2,𝑆9 = 0\µ𝑆𝑛
Š5Đ6´\𝑛= p    q
Aœ1
Bœ4
Cœ5
Dœ4 ā 5
ã2 6-3p2024·Ŕŕ·ĄĚqÏ
 {𝑎𝑛}!3 n 14¸𝑆𝑛\K(𝑛,𝑆𝑛)(𝑛∈N∗)_j
𝑓(𝑥) = 𝐴𝑥2
+𝐵𝑥+ 𝐶(𝐴,𝐵,𝐶∈𝑅)!ÚŢ\µp  q
Aœ𝐶0 = 1
Bœ·𝐴= 0\µ∃𝑛0 ∈N∗\ń𝑆𝑛5–
Cœ·𝐴> 0\µ∃𝑛0 ∈N∗\ń𝑆𝑛5–
Dœ·𝐴< 0\µ∃𝑛0 ∈N∗\ń𝑆𝑛5–
 7  
 !78/
ĕ 7p2024·īţ·ĄĚqŤť!ŦŧŨũŪhūŬŭŮ\ĪŮůŰ{ 25 űâ 36.5 ű!Ŭŭ\ŲųzŴ
ŵ 0.5 ű¸Ŷ C
 œEhŷ\ŤťŸĉŹ!ŦŧŃźh	!Ŭŭ\ƒăĊ(ŻżŲų—L
čŽ\Ťť_čŽ8ŢÒĩÞÖ žŽŲųœ}ŷſ\Ťť!Ŧŧƀ=¦żžŽŲųsŻż\ƁŤťÊë
ƂâÒĩžŽŲų!čŽ8\ŹƃĩõJAh Ųųs 28.5 ű\YÓ_À´óÄEEŽŲųƄ¡!ē4
s 677 űœd_=s\ÜÝh žŽŲųsp    q
Aœ28 ű
Bœ29.5 ű
Cœ32.5 ű
Dœ34 ű
ã2 7-1p2023·Ņņƅę·hĚqƆƇƈ&ŨƉsƊmţġƋ´ƌƍ!
ƌæ\JAE̚
=
ƎE̚J
\±±
zƏĄ\ÛÛ
zƏ±\ƐƐ
zƏĄ\=}ËƑV×s
Ehżƒ\̚
ƓEƔƕ \ƃšƓó£è± ± 
\gƏÑ 2  ŒÛ Û 
\gƏÑ 3  ŒƐ Ɛ 
\ģ
gƏÑ 2  .¨żƒ!
$sƔƕ Ƒ·h ýŃ
Ɩƒ±ƗĄ\ƖƒÛƗ±\ó¨Ƙ!ýŃ
°Đâ–
Ŷ \µ3 5  
!4¸p    q
Aœ189
Bœ190
Cœ191
Dœ192
ã2 7-2p2024·ļĺƙĽ·hĚqƚƛ_x%Î\źƜƝƞ¸ƟƠgơƜƝ\Ƣ‘ƣƤƥĻQ™ÎpĻ
Q³ƦåƧƒq\rhŷƣƤõ 1 ĻQ\ƒſƃLƨƩƣƤ\¢ŷÄõĻQ¤gƪ3hŷƔ 2 Qœ·źŷ
êƣƤ\µÀŷƫEĻQ\ÓrĄŷƣƤƬ{ 1 ĻQƭƮœźgơƯ)ƣƤ™Î\{ 3 ư 1 ƱƭƮ\
â 3 ư 20 ƱŹÙõ 193 ĻQ\AƲEhŷêƣƤ\µŹêƣƤ!¦ŷsp    q
Aœ3 ư 5 Ʊā 3 ư 16 Ʊ
Bœ3 ư 6 Ʊā 3 ư 15 Ʊ
Cœ3 ư 7 Ʊā 3 ư 14 Ʊ
Dœ3 ư 8 Ʊā 3 ư 13 Ʊ
ã2 7-3p2024·Ņņƅę·hĚqƆƇƈ&ŨƉsƊmţġƋ´ƌƍ!
ƌæ\JAE̚
=
ƎE̚J
\±±
zƏĄ\ÛÛ
zƏ±\ƐƐ
zƏĄ.=}ËƑV×s
Ehżƒ\̚Ɠ
EƔƕ \ƃšƓó£è± ± 
\gƏÑ 2  ŒÛ Û 
\gƏÑ 3  ŒƐ Ɛ 
\ģg
      
      
ƏÑ 2  \¨żƒ!
$sƔƕ Ƒ·h ýŃ
Ɩƒ±ƗĄ\ƖƒÛƗ±\ó¨Ƙ!ýŃ
°Đâ–
Ŷ \µ3 10  
!4¸p    q
Aœ754
Bœ755
Cœ756
Dœ757
 8  
 !9:1;<=
ĕ 8p2024·uĤĥƳ4ƴƵ·hĚqéš
 {𝑎𝑛}!3𝑛14¸𝑆𝑛,𝑎1 = 3Ó𝑆𝑛+ 𝑆𝑛+1 = 2𝑛2 +6𝑛+ 3,𝑛∈
N*œ
(1)M𝑆9!6Œ
(2)M
 {𝑎𝑛}!012œ
ã2 8-1p2023·ļŇƶƷ·hĚqéš
 {𝑎𝑛}!Ƹ1œ¸ý
\ĩ𝑆𝑛¸{𝑎𝑛}!3 n 14\Ó2𝑆𝑛= 𝑎𝑛2 +
𝑎𝑛.
(1)M
 {𝑎𝑛}!012Œ
(2)ĩ𝑐𝑛= (−1)𝑛𝑎𝑛𝑎𝑛+1\M
 {𝑐𝑛}!3 n 14𝑇𝑛.
ã2 8-2p2024·īġ·ĚěĜĝq
 {𝑎𝑛}A\𝑎1 = 1\𝑎2 = 9\Ó𝑎𝑛+2 + 𝑎𝑛= 2𝑎𝑛+1 +8\
(1)M
 {𝑎𝑛}!012Œ
(2)
 {𝑏𝑛}!3𝑛14¸𝑆𝑛\ÓĞğ𝑏2
𝑛= 𝑎𝑛\𝑏𝑛𝑏𝑛+1 < 0\M𝑆𝑛œ
ã2 8-3p2024·lm·ĚěĜĝqéš
 {𝑎𝑛}!3𝑛1Ļ¸𝑇𝑛,𝑎𝑛≠0,𝑎𝑛=
𝑇𝑛
𝑇𝑛−2œ
(1)M*

 {𝑇𝑛}s
 \YM
 {𝑎𝑛}!012Œ
      
      
(2)ƹ𝑏𝑛= (−1)𝑛−1(𝑎𝑛+ 1)(𝑎𝑛+1−1)\M
 {𝑏𝑛}!3𝑛14𝑆𝑛œ
 9  >?@6!
 =
ĕ 9p2024·ŅņC¤·ĄĚqéš
 {𝑎𝑛}!3 n 14𝑆𝑛= −
1
2𝑛2 +𝑘𝑛(𝑘∈𝑁*)\Ó𝑆𝑛!5–6¸
9
2œ
(1)Ċ(y
𝑘\YM𝑎𝑛Œ
(2)M
 {|𝑎𝑛|}!3 15 14𝑇15œ
ã2 9-1p2024·ijĴƺĽ·ĄĚqéš
 {𝑎𝑛}s¹1¸ 1 !
 \𝑑> 0\Ï
 {𝑎𝑛}!3𝑛1
4¸𝑆𝑛\Ó𝑆1\𝑆2\𝑆4Cƪ
 œ
(1)M{𝑎𝑛}!012Œ
(2)M
 {|𝑎𝑛−8|}!3𝑛14𝑇𝑛œ
ã2 9-2p2024·lm·ĚěĜĝqéš
 {𝑎𝑛},𝑎1 = −10\ĩ𝑆𝑛¸{𝑎𝑛}!3𝑛14\{Ñƻ①②③
AƼŠh æ¸İı\SeÑƻ=œ①2𝑎5 + 𝑎8 = 0Œ②𝑆11 = −55Œ③
𝑆7
7 −
𝑆5
5 = 2œ
(1)M𝑆𝑛!5Đ6Œ
(2)Ï{|𝑎𝑛|}!3𝑛14¸𝑇𝑛\M𝑇20œ
      
      
ã2 9-3p2024·ĿŇ·ĚěĜĝqéš
 {𝑎𝑛}){𝑏𝑛}¸
 \𝑎2 = 𝑏3\𝑎1 = 2𝑏1\{𝑎𝑛}3𝑛14¸
19𝑛+𝑛2
2
œ
(1)M‘{𝑎𝑛}){𝑏𝑛}!012Œ
(2)sƽĔ_¢h1¤sŃ
!
 {𝑐𝑛}\ńõ@xîV𝑛∈N+\𝑐𝑛¤^Ğğ
|𝑎𝑛+𝑏𝑛|−|𝑎𝑛−𝑏𝑛|
2
≤𝑐𝑛≤
|𝑎𝑛+𝑏𝑛|+|𝑎𝑛−𝑏𝑛|
2
œ·Ĕ_\M‘ÄEŢƾ!{𝑐𝑛}Œ·ÌĔ_\ƿǀ+°œ
 10  
 A!BCD=
ĕ 10p2024·ŀęǁǂǃ·±Ěqéš{𝑎𝑛}¸
 \Ó𝑎5＝3𝑎1\𝑎1 + 𝑎5 + 𝑎14＝𝑎10 +24œ
(1)M{𝑎𝑛}!012Œ
(2)·2𝑛⋅𝜆≥𝑎1 + 𝑎2 +⋯+ 𝑎𝑛BCD\MDŽ
 λ !Š6ċȜ
ã2 10-1p2024·lm·ĚěĜĝqéš
 {𝑎𝑛}!3𝑛14¸𝑆𝑛\Ó{𝑎𝑛+ 𝑆𝑛}sƒ 2 ¸!
 œ
(1)·𝑎1 ≠2\M*
{𝑎𝑛−2}sƪ
 Œ
(2)@îV𝑛,𝑚∈N*,𝑚≠𝑛\¤E
𝑆𝑛−𝑆𝑚
𝑛−𝑚> 1CD\M𝑎1!Š6ċȜ
ã2 10-2p23-24 ±Ţ·ļŇDždž·LJLjqéš𝑆𝑛¸Ƹ1œ¸ý
!
 {𝑎𝑛}!3𝑛14,𝑎1 ∈(0,2),𝑎2
𝑛+3
𝑎𝑛+2 = 6𝑆𝑛.
(1)M{𝑎𝑛}!012Œ
      
      
(2)Ï𝑏𝑛=
1
𝑎𝑛𝑎𝑛+1\
 {𝑏𝑛}!3𝑛14¸𝑇𝑛,·@∀𝑛∈N∗,𝑡≤4𝑇𝑛BCD\MDŽ
𝑡!5–6.
ã2 10-3p23-24 ĄÑ·ljNJNjnj·ŊŋŌōqÏý1
 {𝑎𝑛}!3𝑛1z4𝑏𝑛= 𝑎1 + 𝑎2 +⋯+ 𝑎𝑛\
 {𝑏𝑛}!3𝑛1zĻ𝑐𝑛= 𝑏1𝑏2⋯𝑏𝑛\Ó𝑏𝑛+ 𝑐𝑛= 1.
(1)M*
 1
𝑐𝑛¸
 \YQŐM{𝑎𝑛}､{𝑏𝑛}!012Œ
(2)Ï
 {𝑎𝑛⋅𝑏𝑛+1}!3𝑛14¸𝑆𝑛\Ì2𝑆𝑛>
1
𝜆+𝜆−
13
6 @îVýŃ
𝑛BCD\MýDŽ
𝜆!Š6ċČ.
 11  )
 E!(FGHI=
ĕ 11p2024·řŚĖ·±ĚqžŸ n 1EǍ
 {𝑎𝑛}Ğğ𝑎1 = 𝑎𝑛\𝑎2 = 𝑎𝑛−1\…\𝑎𝑛= 𝑎1\ò𝑎𝑖= 𝑎𝑛−𝑖+1
(𝑖= 1,2,⋯,𝑛)\µǎEǍ
 {𝑎𝑛}¸“@ǎ
 ”.
(1)Ï
 {𝑏𝑛}s1
¸ 7 !“@ǎ
 ”\JA𝑏1,𝑏2,𝑏3,𝑏4C
 \Ó𝑏2 = 3,𝑏5 = 5\û»Ŏ‘
 {𝑏𝑛}!
¢h1Œ
(2)Ï
 {𝑐𝑛}s1
¸2𝑘−1(𝑘∈𝑁∗Ó𝑘≥2)!“@ǎ
 ”\ÓĞğ|𝑐𝑛+1−𝑐𝑛| = 2\ĩ𝑆𝑛¸
 {𝑐𝑛}!3𝑛1
4.
①·𝑐1\𝑐2\…\𝑐𝑘ǏC8ºÁÂ
 \Ó𝑐𝑘= 2023.À𝑘¸Ë6´\𝑆2𝑘−1Šõ5–6?
②·𝑐1 = 2024\Ó𝑆2𝑘−1 = 2024\M𝑘!5Đ6.
ã2 11-1p2024·ǐǑţǒ·ĄĚq·
 {𝑐𝑛}ÙE𝑚(𝑚∈𝑁*,𝑚≥3)1\@îV𝑖(𝑖∈𝑁*,𝑖≤𝑚)¤E𝑐𝑖𝑐𝑚+1−𝑖
= 𝑆p𝑆¸y
\Ó𝑆> 0q\µǎ
 {𝑐𝑛}s𝑆x𝑚!h Ļ@ǎ
 .éš
 {𝑎𝑛}s𝑆x𝑚!h Ļ@
      
      
ǎ
 .
(1)·𝑚= 3\𝑎1 = 1\𝑎2 = 2\M𝑎3!6Œ
(2)éš
 {𝑏𝑛}s¸𝑑(𝑑≠0)!
 \𝑏1 = −11\·𝑚= 10\𝑎𝑛=
𝑏𝑛+2
𝑏𝑛\M𝑑4𝑆!6Œ
(3)·
 {𝑎𝑛}sƸ1œ¸ýŃ
!8ºÁÂ
 \M*
𝑎𝑚
𝑎1 +
𝑎𝑚−1
𝑎2 + ⋅⋅⋅+
𝑎2
𝑎𝑚−1 +
𝑎1
𝑎𝑚<
5𝑆
3 .
ã2 11-2p2024·ĖėţĨ·ĄĚqéš
 {𝑎𝑛}!3 n 14¸𝑆𝑛œ·@¢h 𝑛∈𝑁∗\EÓǓEh 
𝑚∈𝑁∗\ńõ𝑆𝑚≤𝑎𝑛< 𝑆𝑚+1\µǎ{𝑎𝑛}¸“X 
 ”.ĩ𝑏𝑛= 𝑆𝑚+1−𝑎𝑛\𝑛∈𝑁∗\ǎ
 {𝑏𝑛}¸{𝑎𝑛}!“Ɨ1
 ”.
(1)·{𝑎𝑛}!3Ņ1û»¸ 0\1\−1\1\ǔ'Ǖ{𝑎𝑛}sƽ¸“X 
 ”\Yǀ+°Œ
(2)·𝑆𝑛= 2𝑛\*+{𝑎𝑛}¸“X 
 ”\YM£!“Ɨ1
 ”!012Œ
(3)éšý1
 {𝑎𝑛}¸“X 
 ”\Ó{𝑎𝑛}!“Ɨ1
 ”¸
 \*+
𝑆𝑛≤(1 + 2𝑛−2)𝑎1œ
ã2 11-3p2024·ŀę·±ĚqQǖűQRpDifferential CryptanalysisqshǗǖűQRßë\ǘ_0
ćǙǚǖű&ë_Ì¥ǛôQÑǜƚ!Ǜ‘Q\ƢǕ‘ǖű&ë!ǖǝǞ.@x
 {𝑎𝑛}(𝑛∈𝑁*)\
ǟ({Δ𝑎𝑛}¸
 {𝑎𝑛}!hŊQ
 \JAΔ𝑎𝑛= 𝑎𝑛+1−𝑎𝑛Œǟ({Δ2𝑎𝑛}¸{𝑎𝑛}!ĄŊQ
 \JAΔ2𝑎𝑛=
Δ𝑎𝑛+1−Δ𝑎𝑛.žŸ{𝑎𝑛}!hŊQ
 Ğğ|Δ𝑎𝑖| ≠|Δ𝑎𝑗| ∀𝑖,𝑗∈𝑁*,𝑖≠𝑗\µǎ{𝑎𝑛}s“?@Ǡ
 ”ŒžŸ{𝑎𝑛}
!ĄŊQ
 Ğğ|Δ2𝑎𝑖| = |Δ2𝑎𝑗| ∀𝑖,𝑗∈𝑁* \µǎ{𝑎𝑛}s“ǡÌã
 ”.
(1)Ï
 𝐴:1,3,7,9,13,15\'Ǖ
 𝐴sƽ¸“?@Ǡ
 ”ā“ǡÌã
 ”\ƿǀ+°Œ
(2)Ï
 {𝑎𝑛}!012𝑎𝑛= 2𝑛2 +𝑛(𝑛∈𝑁*)\QŐ'Ǖ{Δ𝑎𝑛},{Δ2𝑎𝑛}sƽ¸
 \ƿǀ+°Œ
(3)ÏƸ1œ¸ý
!
 {𝑐𝑛}¸“ǡÌã
 ”\J3𝑛14¸𝑆𝑛\Ó@∀𝑛∈𝑁*\¤E
𝑛
𝑘=1Δ2𝑐𝑘= Δ2
𝑐𝑛+1\@Ğğ𝑛+ 𝑚= 2𝑘(𝑚≠𝑛)!îVýŃ
𝑛,𝑚,𝑘¤E𝑐𝑚≠𝑐𝑛\ÓÌ2𝑆𝑛+ 𝑆𝑚> 𝑡𝑆𝑘BCD\MDŽ
𝑡
!5–6.
      
      
hG8
1œp2024·ĹĺǢǣ·ĚěĜĝqéš
 {𝑎𝑛}Ğğ𝑎2 + 𝑎3 = 14\Ó𝑎4−𝑎2 = 8\µ¹1𝑎1 = p    q
Aœ1
Bœ2
Cœ3
Dœ4
2œp2024·ķ·ĄĚqéš
 {𝑎𝑛}!3𝑛14¸𝑆𝑛\·
𝑎7
𝑎8 = −1\µ𝑆10 = p    q
Aœ𝑆4
Bœ𝑆5
Cœ𝑆6
Dœ𝑆7
3œp2024·ŷǤŝƷ·±Ěqéš
 {𝑎𝑛}¸Ƹ1̸ą!
 \𝑆𝑛¸
 {𝑎𝑛}!3𝑛14\4𝑆𝑛= 𝑎𝑛⋅
𝑎𝑛+1\µ𝑎8!6¸p    q
Aœ4
Bœ8
Cœ12
Dœ16
4œp2024·Ġġǥǃ·±Ěqéš
 {𝑎𝑛}，{𝑏𝑛}œ¸
 \J3𝑛14QŐ¸𝑆𝑛，𝑇𝑛\Ğğ(2𝑛+ 3)𝑆𝑛
= (3𝑛−1)𝑇𝑛\µ
𝑎7+𝑎8+𝑎9
𝑏6+𝑏10 = p   q
Aœ2
Bœ3
Cœ5
Dœ6
5œp2024·Ŕŕ·ĚěĜĝq2024 knjǦ3ǧ\źǢĽǨ@ǩǪǫǬÞh»“ǭƥnjǮ”!ǯǰ™Î\™Î
ǟµžÑ
óhŷuǭÌƕx 800 i!ǩǪŵǭDZDz{ 1 ƭƮû»dz	\dz	^Ǵ 3 ƖƗ 1\ģ^Ǵ 4
ƖƗ 1 !ǩǪ³ƒǵõnjǮ 1 @\ƽµÌ^ǵõnjǮœ·źŷǶŽİı!ǩǪÙE 2000 Ƿ\µǸǹǵõ 1
@njǮ!Ƿ
¸p    q
Aœ167
Bœ168
Cœ169
Dœ170
6œp2024·ļŇǺij·±Ěqéš𝑆𝑛¸
 {𝑎𝑛}!3𝑛14\𝑎1 = −21\𝑆7 = 𝑆15\µ𝑆𝑛!5Đ6¸p    q
Aœ−99
Bœ−100
Cœ−110
Dœ−121
7œp2023·ǻ·ĄĚqéš
 {𝑎𝑛}!3 30 1A9
1!4¸𝐴\:
1!4¸𝐵\Ó𝐵−𝐴= 45\
2𝐴= 𝐵+ 615\µ𝑎𝑛= p    q
Aœ3𝑛−2
Bœ3𝑛−1
Cœ3𝑛+ 1
Dœ3𝑛+ 2
8œp2024·īġ·ĄĚqéš
 {𝑎𝑛}!3 n 14¸𝑆𝑛\Ó𝑆𝑛= 𝑛2 +𝑚\𝑛∈N*\·@xîV!𝑎∈[0,1]\
Ì2
𝑎𝑛
𝑛< 𝑥2−(1 + 𝑎)𝑥−2𝑎2−𝑎+ 2BCD\µDŽ
 x ³^¸p    q
      
      
Aœ−2
Bœ0
Cœ1
Dœ2
ĄGƔ
9œp2024·řŚĖśŜŝ·ĄĚqéš
 {𝑎𝑛}!¹1𝑎1 = 1\𝑑= 6\_{𝑎𝑛}A¢füÖ1zŴ¤Ǽ
ô𝑘 
\ń£è4ǽ
 !
h¡ǏCh !
 {𝑏𝑛}\Ñ ǀëýĊ!Ep     q
Aœ𝑎𝑛= 6𝑛−5
BœÀ𝑘= 2´\𝑏𝑛= 2𝑛−1
CœÀ𝑘= 2´\𝑏19Ìs
 {𝑎𝑛}A!1
Dœ·𝑏8s
 {𝑎𝑛}A!1\µ𝑘!6³^¸ 6
10œp2024·Ġġǥǃ·ĚěĜĝqéš
 {𝑎𝑛}s¸𝑑(𝑑≠0)!
 \·£!32𝑚(𝑚> 1)1!4
𝑆2𝑚= 0\µÑ ì<ýĊ!sp    q
Aœ·𝑑< 0\ń𝑎𝑛> 0!5–𝑛!6¸𝑚
Bœ𝑆𝑚s𝑆𝑛!5Đ6
Cœ3𝑎2
𝑚+ 𝑎2
𝑚+2 = 𝑎2
𝑚−1 +3𝑎2
𝑚+1
Dœ𝑎2
𝑚−1 + 𝑎2
𝑚= 𝑎2
𝑚+1 + 𝑎2
𝑚+2
11œp2024·ǻ·ĚěĜĝqÏ{𝑎𝑛}sƸ1¸ý!ÊǍ
 \·@x∀𝑛∈𝑁*\𝑎2
𝑛+1−𝑎2
𝑛= 𝑑pd
¸ĸąy

q\µǎ
 {𝑎𝑛}¸ß
 œ¦§p   q
Aœ·{𝑎𝑛}sß
 \µ 𝑎2
n s
 
Bœ
 {2n}¸ß
 
Cœ·{𝑎𝑛}sß
 \µ
 {𝑎𝑛+1−𝑎𝑛}AĔ_Đx 1 !1
Dœ·{𝑎𝑛}sß
 \µĔ_ýŃ
 n\ńõ
𝑛
𝑖=1
1
𝑎𝑖> 2024
±G…†
12œp2024·lm·ĚěĜĝq_
 {𝑎𝑛}A\𝑎1 + 𝑎3 + 𝑎5 = 9\𝑎6 = 9\µJ𝑑=      
.
13œp2024·ĖėǾǿ·±Ěq⌈𝑥⌉(𝑥∈R)®¯ÌĐx x !5ĐŃ
\ĕž⌈2⌉= 2\⌈−3
2⌉= −1œéš
 
{𝑎𝑛}!3 n 14¸𝑆𝑛\Ó𝑆7 = −7\𝑎4 + 𝑎6 = −3œĩ𝑏𝑛= ⌈𝑎𝑛⌉\µ
 {𝑏𝑛}!3 10 1!4¸      
œ
14œp2024·ĖĺȀnj·ĚěĜĝqéš
 {𝑎𝑛}s
 \𝑏𝑛=
𝑎𝑛−8,𝑛为奇数
2𝑎𝑛+ 1,𝑛为偶数
 
\ĩ𝑆𝑛\𝑇𝑛QŐ¸{𝑎𝑛}\
{𝑏𝑛}!3𝑛14\·𝑆3 = 18\𝑇2 = 10\µ𝑇20 =      
œ
ŅGS’
15œp2024·ļĺƙA·ĚěĜĝqéš
 {𝑎𝑛}!3𝑛14¸𝑆𝑛\𝑎1 =
1
2\ÓÀ𝑛≥2(𝑛∈N∗)´\2𝑆𝑛⋅𝑆𝑛−1 = −
𝑎𝑛\
      
      
(1)*+

 1
𝑆𝑛s
 Œ
(2)Ï
 {𝑏𝑛}Ğğ𝑏𝑛=
𝑎𝑛
𝑎𝑛+1\M𝑏1 + 𝑏101!6.
16œp2024·Ņņ·ĚěĜĝqéš
 {𝑎𝑛}Ğğ𝑎1 =
3
2\𝑎𝑛+1 +
1
𝑎𝑛= 2œ
(1)*+
 
1
𝑎𝑛−1 s
 \YM{𝑎𝑛}!012Œ
(2)·
 {𝑏𝑛}Ğğ\𝑏𝑛= (𝑎𝑛−1)(𝑎𝑛+1−1)\M{𝑏𝑛}!3 n 14𝑆𝑛œ
17œp2024·īţ·ĚěĜĝqéš̸ 0 !
 {𝑎𝑛}Ğğ𝑎2
3 = 𝑎1𝑎13\Ó𝑎2𝑎3−𝑎5 = 𝑎4−𝑎1.
(1)M{𝑎𝑛}!012Œ
(2)ĩ𝑆𝑛s
 {𝑎𝑛}!3𝑛14\*+: 
1
𝑆1 +
1
𝑆2 +
1
𝑆3 + ... +
1
𝑆𝑛< 2.
18œp2024·lm·ĚěĜĝq
 {𝑎𝑛}!3𝑛14¸𝑆𝑛\2𝑆2
𝑛−2𝑎𝑛𝑆𝑛+ 𝑎𝑛= 0(𝑛≥2,𝑛∈N∗)\Ó𝑎1 = 1.
(1)*+
 1
𝑆𝑛¸
 Œ
(2)@xîV𝑛∈𝑁*\Ì22𝑛𝑎𝑛+1 +
𝜆
2𝑛−1 < 0BCD\MDŽ
𝜆!Š6ċČ.
      
      
19œp2024·Ėĺţȁ·±Ěqł(
 {𝐴𝑛}\·@îV m\𝑛∈𝑁*Ó𝑚≠𝑛\𝐴𝑚+ 𝐴𝑛s{𝐴𝑛}A!1\µǎ{𝐴𝑛
}¸“H 
 ”œÏ
 {𝑎𝑛}!3 n 14¸𝑆𝑛.
(1)·𝑆𝑛= 𝑛2 +𝑛\ǔ'Ǖ
 {𝑎𝑛}sƽ¸“H 
 ”\Yǀ+°Œ
(2)Ï{𝑎𝑛}IJs
 Ȃs“H 
 ”\Ó𝑎1 = 6\𝑎2 ∈N∗\𝑎2 > 6\M d !ÄE³^6Œ
(3)Ï{𝑎𝑛}s
 \Ó@îV𝑛∈𝑁*\𝑆𝑛s{𝑎𝑛}A!1\M*
{𝑎𝑛}s“H 
 ”œ