专题6.4数列的通项公式的求法十二大题型（原卷版）_11.4-2_2026-11-04

      
      
专题6.4 数列的通项公式的求法【十二大题型】

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 3  # an $ Sn %&'() ......................................................................................................................5
 4  *+  ....................................................................................................................................................5
 5  *,  ....................................................................................................................................................6
 6  -.  ....................................................................................................................................................7
 7  #/0
1%()2'
1() ....................................................................................................8
 8  #/3
1%()2'
1() ....................................................................................................9
 9  45
1%()6 ..........................................................................................................................10
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19
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CDE
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(1)IJ
1%()2
KLM&
(2)NO'
1%()
2%PQ 
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V
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2023 R I U
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12 G
2023 RⅡU
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2023 RSTXUY
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2024 RSTXUY
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1%()2
%'J[f%]AWgBhJi
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rstuvwxyzmno{9|
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1%()2
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Š‹
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1%()2.
      
      
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1%LM2
(1)LM2%œ
Š‹k“
1%žŸ )¡)Ž%&’hk“2”d•–W—˜™“2”¢š›™“
1%
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1LM2%J
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1¥w()2”kƒW¦t[¤w%
1¥wLM2.
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1%k‚Q .©ª«WLM2K()2kƒW¥[¬)%	 n %­/
2.
Š‹®€B'%7¯
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1%µ).
¶LM2'mk“
1W·¸¨m
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1{
}%V 1 )(¡»b))v
LM&——
1{
}%¼½k)
$Œ%»k)
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)(¡»b))%&W¦¾™“&’
h/2d•–.
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1%()2%P¿' 
1‰ 
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1»ÁÂ)W'Ã
1%()xWkl¤¨%)GHWÄÅÆÇW`ÈÉÊÆÇËmÌ

1%k“().
2‰!" 
À‡
1%()2%ÍW¬ÎÏ%()2WÉÊ
1%!"ЀÀ‡ÑÒW'm()2q
%Ï
W`ÈÓÔÌ
1%().
3‰2 
# an $ Sn %&'()
(1)À‡ Sn ' an %PQ [Õ
=
Ö׬ an %&2WØ'()2.
(2) Sn $ an &6%'JÙÚ
QÛ 1
Õ an= Sn -Sn-1(nÜ2)Ö×ÝÎ SnWSn-1 %&2WØ'J.
QÛ 2
Õ Sn -Sn-1= an(nÜ2)Ö×ÝÎ anWan-1 %&2WØ'J.
4‰*+ 
jŠ an+1=an+f(n)%LM&2Õ*+ 'KW?Þ½ßಁá)Wâãá).
5‰*, 
j Š an+1=an · f(n) % L M  & 2 ’  ×
% j 2 W ’  * , W y ’ 
åæ'm().
6‰-. 
£jŠ an+1=pan+q %LM&2’h×(an+1+x)=p(an+x)%j2W-ç%/3
1W'm()2W
'èé x [ê.
§jŠ an+1=pan+qn+c %
1WëæÏ
 xWyW-.%/3
1{
}.
      
      
¶jŠ an+1=pan+qn %
1W ìíîh qn+1W-.%
1{
}.
ïjŠ
(A,B,C ×P
)%
1W’(ð ìíxñò
%Q -.
1'J.
7‰/0
1%()2 
(1)Š‹¨!%
1[/0
1W'mó)K0WôõÕ/0
1%()2'Jv
(2)Š‹¨!%
1’h-.m/0
1Wö'm-.%/0
1%()2Wo(ðLM&2÷ø
èjÖWÓÔ¤'
1%()2.
8‰/3
1%()2 
(1)Š‹¨!%
1[/3
1W'mó)K3WôõÕ/3
1%()2'Jv
(2)Š‹¨!%
1’h-.m/3
1Wö'm-.%/3
1%()2Wo(ðLM&2÷ø
èjÖWÓÔ¤'
1%()2.
 1   
ù 1Y24-25 ú«·ST·ûüýþZ
11,−2
2 ,
1
2,−2
4 ,
1
4,⋯%k“()2×Y    Z
A‰ −1
2
𝑛−1
B‰ −
2
2
𝑛
C‰(−1)𝑛
2
2
𝑛−1
D‰(−1)𝑛+1
2
2
𝑛−1
è2 1-1Y2024·ÿĀ·āĂZuă
1WdĄ¬ąĆćĈĉqĊċ“uăŽ
Čč”%M=WgB¬
JĎqTċDq%ďĐăđĒW
1q%ēk)W¥å•ďĐăđðĔqWĕĖĖėð% Ę
é
ęKW[qĚċDqěĜĝ%Ğğ
Ġ«Vkġ
1‰Ģ» 10 )°±[ 0W2W4W8W12W18W
24W32W40W50WģÌ
1%V 25 )$V 24 )%0×Y    Z
A‰22
B‰24
C‰25
D‰26
è2 1-2Y23-24 ú«·ĤĥĦħ·ĨĩĪīZ
1−2,4,−
26
3 ,20,⋯%k“()2’h[Y    Z
A‰𝑎𝑛= (−1)𝑛⋅2𝑛
B‰𝑎𝑛= (−1)𝑛⋅3𝑛−𝑛
𝑛
C‰𝑎𝑛= (−1)𝑛⋅2𝑛+1−2
𝑛
D‰𝑎𝑛= (−1)𝑛⋅3𝑛−1
𝑛
è2 1-3Y2024·ST·ĂĬĭĮZį» 6 ĞİWıIJ%ijĴĵĶķĸĹĺ
%œxWPPĻ
ļ
ĽçľĿ«%pŀ”WŒŁ÷øµ2µƒ%ł1KGÍWšý“j
”‰ 3 Ńŀ”’hńçk“7āŅjW
íƒ 6 Ńŀ”¡ņ 10 Ńŀ”’hńçŇu%āŅj‰ijĴĵĶķĸĻ 1W3,6,10/šý“āŅ
”¡“ā
Ņj
”‰íxňŁʼnńmI7Qj
9Čìj
9Ŋìj
KĢňìj
‰Šŋ¤–Ōńm%Ŋìj
W
—˜V 20 “Ŋìj
×Y    Z
      
      
A‰778
B‰779
C‰780
D‰781
 2  !" 
ù 2Y23-24 úō·ST·ûüýþZÀ‡
1{𝑎𝑛}qW𝑎1 = 3W𝑎10 = 21W()𝑎𝑛[)
𝑛%k±~
W
(1)'{𝑎𝑛}%()2W¦'𝑎2000v
(2)Á{𝑏𝑛}[#𝑎2,𝑎4,𝑎6,𝑎8,⋯,ŎçWŏŐő{𝑏𝑛}%k“()2‰
è2 2-1Y23-24 ú«·Œœ4Ŕ·ĨĩĪīZo
1{𝑎𝑛}qWÀ‡𝑎𝑛=
𝑎𝑛
𝑏𝑛+1W¾𝑎2 =
6
5,𝑎3=
9
7.
(1)'()2𝑎𝑛.
(2)'ŕ
{𝑎𝑛}[LŖ
1.
è2 2-2Y23-24 āō·ŗ·ĨĩĪīZÀ‡𝑓(𝑥)[
~
¾ŋŘðA
5, 1
2 W
1{𝑎𝑛}řŚ𝑎𝑛= 𝑓
1−
1
𝑛+1 (𝑛∈N∗)‰
(1)'
1{𝑎𝑛}%()2v
(2)ś
1{𝑎𝑛}%» n )K×𝑆𝑛WÁ|𝑆𝑚| ≥2(𝑚∈N∗)W' m %Ŝpŝ‰
      
      
è2 2-3Y24-25 ú«·ST·ûüýþZ!"
1“`Vú)ŞWÁ
1{𝑎𝑛}%ēk)$»k)%şQ
0×íkP
 dWģŠ
1{𝑎𝑛}×/şQ0
1Wd šýÌ
1%şQ0‰”À‡
1{𝑎𝑛}ד/şQ0
1”W¾𝑎1 = 1W𝑎5 = 3‰
(1)šŢřŚÑÒ%
1{𝑎𝑛}[ţŤkW¦ťŦ#v
(2)'7)
1{𝑎𝑛}%()2W¦šŢĢŧŨũ‰
 3  # an $ Sn %&'()
ù 3Y2024·Ūū·ĂĬĭĮZÀ‡
1{𝑎𝑛}%»𝑛)K×𝑆𝑛WÁ𝑆𝑛= 2𝑛−1−
1
2Wģ
1{𝑎𝑛}%()2×
Y    Z
A‰𝑎𝑛=
1
2 ,𝑛= 1,
2𝑛,𝑛≥2
 
B‰𝑎𝑛= 2𝑛−1
C‰𝑎𝑛= (−2)𝑛−2
D‰𝑎𝑛= 2𝑛−2
è2 3-1Y23-24 úō·ŬŭuŮ·5qZÀ‡
1{𝑎𝑛}%»𝑛)K𝑆𝑛= 𝑛2 +1Wģ
1{𝑎𝑛}%()2×
Y    Z
A‰𝑎𝑛= 𝑛+ 1
B‰𝑎𝑛= 2𝑛−1
C‰𝑎𝑛= 2𝑛+ 1
D‰𝑎𝑛=
2 ,  𝑛= 1 ,  
 2𝑛−1,  𝑛≥2
 
è2 3-2Y2024·ůĥ·ĂĬĭĮZÀ‡
1{𝑎𝑛}řŚ
𝑛
𝑘=1
𝑎𝑘
2𝑘−1 = 𝑛+ 1Wģ𝑎2024 = Y    Z
A‰2024
B‰2023
C‰4047
D‰4048
è2 3-3Y2024·Ūū·āĂZÀ‡
1{𝑎𝑛}řŚ2𝑎1 + 22𝑎2 + 23𝑎3 + ⋅⋅⋅+ 2𝑛𝑎𝑛= 𝑛⋅2𝑛Wģ{𝑎𝑛}%()
2×Y    Z
A‰𝑎𝑛=
1,𝑛= 1
𝑛+ 1,𝑛≥2
 
B‰𝑎𝑛=
𝑛+1
2
C‰𝑎𝑛= 𝑛
D‰𝑎𝑛=
1,𝑛= 1
𝑛−1,𝑛≥2
 
 4  *+ 
ù 4Y23-24 úō·ŗŰűŲų·ŴŏZo
1{𝑎𝑛}qW𝑎1 = 1W𝑎𝑛+1 = 𝑎𝑛+
1
𝑛(𝑛+1)Wģ𝑎𝑛/¬
Y    Z
      
      
A‰
1
𝑛
B‰
2𝑛−1
𝑛
C‰
𝑛−1
𝑛
D‰
1
2𝑛
è2 4-1Y23-24 ú«·Ŭŭ·ĨĩĪīZo
1{𝑎𝑛}qW𝑎1 = 2,𝑎𝑛+1 = 𝑎𝑛+ ln 1 + 1
𝑛Wģ𝑎𝑛= Y    Z
A‰2 + 𝑛ln𝑛
B‰2 + (𝑛−1)ln𝑛
C‰2 + ln𝑛
D‰2 + 𝑛+ ln𝑛
è2 4-2Y2024·ŵœŶŒ·kĂZÀ‡
1{𝑎𝑛}řŚ
𝑎1 = 9,𝑎𝑛+1−𝑎𝑛= 2𝑛Wģ𝑎4 = Y    Z
A‰21
B‰23
C‰25
D‰27
è2 4-3Y23-24 ú«·ŷŸŹź·5ŻZċťżıIJijĴĵĶķĸ%
Žľž¡pŀ”dĹĺ

‰ňŁÉÊľž¡pŀſ¤ł1%jƀĻ
GçƁÍWŠŋ% 1W5W12W22 Š×Čìj
WÁČìj

¤-ç%
1śý{𝑎𝑛}Wō1t[
1{𝑎𝑛}%)%[Y    Z
A‰35
B‰70
C‰145
D‰170
 5  *, 
ù 5Y23-24 ú«·\ƂƃƄƅ·5ŻZÀ‡𝑎1 = 2W𝑎𝑛= 𝑛(𝑎𝑛+1−𝑎𝑛)Wģ
1{𝑎𝑛}%()2[𝑎𝑛=
YƆƆZ
A‰n
B‰𝑛+ 1
C‰2n
D‰ 𝑛+1
𝑛
𝑛
è2 5-1Y23-24 úō·Œœ·5qZÀ‡
1{𝑎𝑛}řŚ𝑎1 =
1
3W𝑎𝑛=
2𝑛−3
2𝑛+1𝑎𝑛−1(𝑛≥2W𝑛∈𝑁∗)Wģ
1{𝑎𝑛}
%()𝑎𝑛= Y    Z
A‰
1
4𝑛2−1
B‰
1
2𝑛2+1
C‰
1
(2𝑛−1)(2𝑛+3)
D‰
1
(𝑛+1)(𝑛+3)
è2 5-2Y2024·ÿĀƇƈ·kĂZƉ
1{𝑎𝑛}%» n )K×𝑆𝑛W¾𝑎1 = 1W{𝑆𝑛+ 𝑛𝑎𝑛}×P
1Wģ𝑎𝑛=
Y    Z
A‰
1
3𝑛−1
B‰
2
𝑛(𝑛+1)
C‰
6
(𝑛+1)(𝑛+2)
D‰
5−2𝑛
3
è2 5-3Y23-24 āō·ST·ĨĩĪīZÀ‡
1{𝑎𝑛}%»𝑛)K×𝑆𝑛W𝑎2 = 4W𝑆𝑛=
(𝑛+1)𝑎𝑛
2
(𝑛∈𝑁∗)W
      
      
ģ
1{𝑎𝑛}%()2×Y    Z
A‰𝑎𝑛= 2𝑛(𝑛∈𝑁∗)
B‰𝑎𝑛= 2𝑛(𝑛∈𝑁∗)
C‰𝑎𝑛= 𝑛+ 2(𝑛∈𝑁∗)
D‰𝑎𝑛= 𝑛2(𝑛∈𝑁∗)
 6  -. 
ù 6Y2024·ST·ĂĬĭĮZÀ‡
1{𝑎𝑛}%» n )K𝑆𝑛= 2𝑎𝑛−𝑛‰
(1)'{𝑎𝑛}%()2v
(2)ŕŦ
𝑎1+1
𝑎2 +
𝑎2+1
𝑎4 +
𝑎3+1
𝑎6 +⋯+
𝑎𝑛+1
𝑎2𝑛<
5
4‰
è2 6-1Y2024·ůĥĥƊ·kĂZÀ‡
1{𝑎𝑛}%»𝑛)K×𝑆𝑛W𝑎1 = 1W¾řŚ(𝑛+ 1)𝑆𝑛= 𝑛𝑆𝑛+1−
1
2𝑛
(𝑛+ 1).
(1)'
1{𝑎𝑛}%()2v
(2)Ɖ𝑏𝑛= 𝑎2
𝑛+ 3𝑎𝑛⋅cos𝑛πW'
1{𝑏𝑛}%»𝑛)K𝑇𝑛.
è2 6-2Y2024 āō·Ūū祷ĪīZÀ‡
1{𝑎𝑛}%»𝑛)K×𝑆𝑛W¾řŚ𝑆𝑛= 2𝑎𝑛+2𝑛−1‰
(1)'ŕ

1{𝑎𝑛−2}×/3
1v
(2)À‡𝑏𝑛=
𝑛(2−𝑎𝑛)
3
W'
1{𝑏𝑛}%»𝑛)K‰
è2 6-3Y2024·ST·ĂĬĭĮZÀ‡
1{𝑎𝑛}%»𝑛)K×𝑆𝑛W¾2𝑆𝑛= (𝑛+ 2)(𝑎𝑛+ 1)‰
(1)'
1{𝑎𝑛}%()2v
      
      
(2)'ŕ
−
3
4 <
1
𝑆1 +
1
𝑆2 + ⋅⋅⋅+
1
𝑆𝑛≤−
1
3‰
 7  #/0
1%()2'
1()
ù 7Y23-24 ā«·Ƌƌƍū·ĨĩĪīZÀ‡
1{𝑎𝑛}řŚ𝑎1 = 1W¾A(
1
𝑎𝑛+1,
1
𝑎𝑛)oôƎ𝑦= 𝑥−2«.
(1)'
1{𝑎𝑛}%()2v
(2)
1{𝑎𝑛𝑎𝑛+1}»𝑛)K×𝑇𝑛W'ßƏ𝑇𝑛< 3𝑚−12𝑛∈𝑁*­çƐ%𝑚Y𝑚∈ZZ%Ŝpŝ.
è2 7-1Y2024·ST·ĂĬĭĮZÀ‡
1{𝑎𝑛}%»𝑛)K×𝑆𝑛W¾𝑎1 = −2W𝑆𝑛+1 + 𝑆𝑛= 𝑎𝑛+1 +2𝑛𝑎𝑛−𝑛2
+𝑛W𝑛∈N*‰
(1)'
1{𝑎𝑛}%()2v
(2)Á𝑎𝑛+1 ≤2(𝑘⋅3𝑎𝑛+3−𝑎𝑛)¼½%𝑛∈N*­çƐW'ª
𝑘%Ŝpŝ‰
è2 7-2Y2024·ŷŸƑŮ·ĂĬĭĮZÀ‡/0
1{𝑎𝑛}řŚ𝑎6 + 𝑎7 = 4W¾𝑎1,𝑎4,𝑎5ç/3
1‰
(1)'{𝑎𝑛}%()2v
(2)ś𝑇𝑛×
1{𝑎𝑛}»𝑛)%,ƒWÁ𝑎1 < 0W'𝑇𝑛%Ŝuŝ‰
è2 7-3Y2023·Œœ·āĂZÀ‡
1{𝑎𝑛}%» n )K×𝑆𝑛W𝑎1 = 1W2𝑛𝑆𝑛+1−2(𝑛+ 1)𝑆𝑛= 𝑛(𝑛+ 1)‰
      
      
(1)'
1{𝑎𝑛}%()𝑎𝑛v
(2)Ɖ𝑏𝑛=
𝑎𝑛+2
2𝑛+2⋅𝑆𝑛W'
1{𝑏𝑛}%» n )K𝑇𝑛‰
 8  #/3
1%()2'
1()
ù 8Y2024·ST·ĂĬĭĮZÀ‡
1{𝑎𝑛}%»𝑛)K×𝑆𝑛W¾𝑆𝑛=
3
4 +
3(2𝑛−1)
4𝑛
𝑎𝑛(𝑛∈𝑁∗)‰
(1)'
1{𝑎𝑛}%()2v
(2)Ɖ𝑏𝑛=
3𝑛
(𝑛+2)𝑎𝑛W
1{𝑏𝑛}%»𝑛)K×𝑇𝑛W'ŕ
𝑇𝑛<
3
4‰
è2 8-1Y2024·ST·ĂĬĭĮZś𝑆𝑛×
1{𝑎𝑛}%» n )KWÀ‡𝑎1 = 1,{2𝑎𝑛−𝑆𝑛}[0× 1 %/0

1.
(1)ŕŦ
1{𝑎𝑛+ 1}[/3
1W¦'{𝑎𝑛}%()2v
(2)Á𝑏𝑛=
𝑛(𝑛+1)
𝑆𝑛+𝑛+2W𝑏𝑘[
1{𝑏𝑛}%Ŝu)W'7¯
 k %ŝ.
è2 8-2Y2024·ST·ĂĬĭĮZÀ‡
1{𝑎𝑛}%ó)𝑎1 = 1W¾řŚ𝑎𝑛+1 + 𝑎𝑛= 3𝑛+ 1.
(1)ŕŦ 𝑎𝑛−3
2 𝑛+ 1
4 [/3
1W¦'
1{𝑎𝑛}%()2v
(2)[ţƓo7¯
𝑚WƏӐ¼½%7¯
𝑛W𝑎𝑚+ 𝑎𝑛= 𝑎𝑚+𝑛ęçƐƔÁƓoW'm𝑚%ŝvÁtƓoW
ƕťŦ#.
      
      
è2 8-3Y2024·ŸĥœƖ·úĂZÀ‡
1{𝑎𝑛}%»𝑛)K×𝑆𝑛W¾řŚ𝑎1 = 1,𝑎2 = 3,𝑎𝑛+ 𝑎𝑛+2 = 𝑘
𝑎𝑛+1.
(1)Ɨ𝑘= 2xW'𝑆10;
(2)Á𝑘=
5
2WƉ𝑏𝑛= 𝑎𝑛+1−2𝑎𝑛W'{𝑏𝑛}%()2.
 9  45
1%()6
ù 9Y2024 ā·ST·ĪīZÀ‡
1{𝑎𝑛}řŚ𝑎1 = 2W𝑎𝑛𝑎𝑛+1 + 𝑎𝑛−𝑎𝑛+1 +1 = 0Wś
1{𝑎𝑛}%
» n )K×𝑆𝑛W» n )ƒ×𝑇𝑛WģY    Z
A‰
1{𝑎𝑛}[45
1
B‰𝑎2024 =
1
3
C‰𝑆2024 > 𝑇2024
D‰𝑇2024 = 1
è2 9-1Y23-24 úō·ĤƘƙƚ·5qZ
1{𝑎𝑛}%» n )K×𝑆𝑛W¾řŚ𝑎1 = 1W𝑎𝑛+1 =
2𝑎𝑛,𝑛是奇数
1
𝑎𝑛,𝑛是偶数
 
Wģō1ť 7ƛ%wY    Z
A‰𝑎3 =
1
2
B‰{𝑎𝑛}[45
1
C‰𝑎2022 = 2
D‰𝑆18 = 20
è2 9-2Y23-24 ā«·ĤƘƜƝ·ĨĩĪīZÀ‡~
𝑓(𝑥) =
𝑥+ 1
2 ,𝑥≤1
2
2𝑥−1, 1
2 < 𝑥< 1
𝑥−1,𝑥≥1
 
WÁ
1{𝑎𝑛}řŚ𝑎1 =
7
3W𝑎𝑛+1 = 𝑓(𝑎𝑛)W𝑛∈N∗ģō1ť 7ƛ%[Y    Z
A‰Ã
1[45
1¾45× 3
B‰Ã
1t[45
1
C‰𝑎2023 + 𝑎2024 =
3
2 
D‰𝑎2023 + 𝑎2024 =
7
6
è2 9-3Y2024·\ƂƇƞ·ĂĬĭĮZÀ‡𝑆𝑛[{𝑎𝑛}%»𝑛)K𝑎1 = 2W𝑎𝑛= 1−
1
𝑎𝑛−1，𝑛≥2,𝑛∈N∗Wģō
1)ƟƠ%[Y    Z
      
      
A‰𝑎2021 = 2
B‰𝑆2021 = 1012
C‰𝑎3𝑛𝑎3𝑛+1𝑎3𝑛+2 = 1
D‰{𝑎𝑛}[h3×45%45
1
 10  789:;<='()
ù 10Y2024·^ơżƢKƣ?·kĂZÀ‡
1{𝑎𝑛}%»𝑛)K×𝑆𝑛,𝑎1 = 3¾𝑆𝑛+ 𝑆𝑛+1 = 2𝑛2 +6𝑛+ 3,𝑛∈
N*‰
(1)'𝑆9%ŝv
(2)'
1{𝑎𝑛}%()2‰
è2 10-1Y2024·ŒŬƤź·āĂZÀ‡
1{𝑎𝑛}řŚ
𝑎𝑛𝑎𝑛+1
2
= 4𝑛W𝑎1 = 2W𝑛∈𝑁∗.
(1)'
1{𝑎𝑛}%()2v
(2)Ɖ𝑏𝑛=
𝑎𝑛−1
𝑎𝑛+1W
1{𝑏𝑛}%»𝑛)K×𝑆𝑛W'ŕ
𝑛−2 < 𝑆𝑛< 𝑛.
è2 10-2Y2024·ůĥƊƥ·ĂĬĭĮZś𝑆𝑛×
1{𝑎𝑛}%»𝑛)KWÀ‡𝑎1 = 1,𝑛𝑆𝑛+1−(𝑛+ 1)𝑆𝑛= 𝑛2
+𝑛.
(1)'{𝑎𝑛}%()2v
(2)Á𝑏𝑛= (−1)𝑛𝑎𝑛+ [(−1)𝑛+ 1]2𝑛W'
1{𝑏𝑛}%»2𝑛)K𝑇2𝑛.
è2 10-3Y2024·ƦœƇľ·āĂZÁµ)Ƨ×7
%
1{𝑐𝑛}řŚ𝑐𝑛𝑐𝑛+2−𝑐2
𝑛+1 = 𝑘𝑐𝑛𝑐𝑛+1Y𝑛∈𝑁*,𝑘×
P
ZWģŠ{𝑐𝑛}ד30/
1”.À‡{𝑎𝑛}ד30/
1”W¾𝑎1 =
5
8,𝑎2 =
15
16,3𝑎4 = 2𝑎5.
      
      
(1)'{𝑎𝑛}%()2v
(2)Ɖ𝑏𝑛=
𝑎𝑛,𝑛为奇数
𝑏𝑛−1 + 1,𝑛为偶数
 
W'
1{𝑏𝑛}%»𝑛)K𝑆𝑛.
 11  >
1%()6
ù 11Y2024·\ƂƃƄƅ·āĂZÀ‡𝑆𝑛[/0
1{𝑎𝑛}%»𝑛)KW𝑆5 = 𝑎11 = 20W
1{𝑏𝑛}[3u¬
1 %/3
1W¾𝑏2
3 = 𝑏6W𝑏4−𝑏2 = 12.
(1)'
1{𝑎𝑛}K{𝑏𝑛}%()2v
(2)Ɖ𝑐𝑛=
𝑆𝑛
𝑏𝑛W'Ə𝑐𝑛ñÓŜuŝx𝑛%ŝ.
è2 11-1Y2024·ƨƄŸƩƪƫ·ĂĬĭĮZÀ‡
1{𝑎𝑛}%» n )ƒ×𝑇𝑛= 3
𝑛(𝑛−1)
2
W
1{𝑏𝑛}řŚ𝑏1
= 1W 𝑏𝑛−𝑏𝑛−1 = 1Y𝑛≥2W𝑛∈𝑁∗Z‰
(1)'
1{𝑎𝑛}W{𝑏𝑛}%()2v
(2)Ƭ
1{𝑎𝑛}W{𝑏𝑛}q%ƭ)`pÔuł1-ç
1{𝑐𝑛}W'
1{𝑐𝑛}%()2‰
è2 11-2Y2024·ŪūƮƯ·āĂZÀ‡{𝑎𝑛}[/0
1W{𝑏𝑛}[/3
1W¾{𝑏𝑛}%»𝑛)K×𝑆𝑛W2𝑎1 =
𝑏1 = 2W𝑎5 = 5(𝑎4−𝑎3)Wo①𝑏5 = 4(𝑏4−𝑏3)W②𝑏𝑛+1 = 𝑆𝑛+2™ “ÑÒq¼Ģqk“WưçōƱ6
%Ji.
(1)'
1{𝑎𝑛}K{𝑏𝑛}%()2v
      
      
(2)Ɖ
1 𝑎𝑛
𝑏𝑛%»𝑛)K×𝑇𝑛W'𝑇𝑛.
è2 11-3Y2024·ƲƳŬƴ·āĂZÀ‡{𝑎𝑛}×/0
1W»𝑛)K×𝑆𝑛WÁ𝑎2 = 3W𝑆8 = 6𝑆3 +10v
1
{𝑏𝑛}řŚ
 1−1
𝑏1
1−1
𝑏2 ⋯1−1
𝑏𝑛=
1
𝑏𝑛W𝑛∈𝑁∗.
(1)'{𝑎𝑛}K{𝑏𝑛}%()2v
(2)¼½%𝑚∈𝑁∗WƬ{𝑎𝑛}qƵæƶ(2𝑚,22𝑚)^)%“
ś×{𝑐𝑚}.
YiZ'𝑐𝑚v
YiiZś𝑑𝑚=
2
2𝑏2(𝑚−1)−𝑐𝑚W{𝑑𝑚}%»𝑚)Kś×𝑇𝑚W[ţƓo𝑚W𝑡∈𝑁∗WƏÓ
𝑇𝑚+1−𝑡
𝑇𝑚−𝑡= 𝑑𝑡+1çƐƔÁƓoW
'm𝑚𝑡%ŝvÁtƓoWƕťŦ#.
 12  ?@
1'()
ù 12Y2024·ƷźƷƯ·āĂZÀ‡7)
1{𝑎𝑛}%»𝑛)K×𝑆𝑛W¾řŚ𝑎1 = 1,𝑆𝑛=
𝑎𝑛𝑎𝑛+1
2
.ŏ':
(1)
1{𝑎𝑛}%()2;
(2)ś𝑐𝑛= 𝑎2𝑛W
1
1
𝑐𝑛𝑐𝑛+1 %»𝑛)K×𝑇𝑛WƗ𝑇𝑛>
2
9xW'řŚÑÒ%Ŝp¯
𝑛.
è2 12-1Y2024·ůĥƸœ·ĂĬĭĮZÀ‡µ)Ƨ×7
%
1{𝑎𝑛}%»𝑛)K×𝑆𝑛,2𝑆𝑛= 𝑛(𝑎𝑛+ 1)¾
𝑎2 =
3
2𝑎1.
(1)'{𝑎𝑛}%()2v
      
      
(2)Á𝑏𝑛=
𝑎𝑛
2𝑛,'
1{𝑏𝑛}%»𝑛)K𝑇𝑛.
è2 12-2Y2024·Ĥĥ·ĂĬĭĮZÀ‡
1{𝑎𝑛}%»𝑛)K×𝑆𝑛W¾
1
𝑆1 +
1
𝑆2 + ⋅⋅⋅+
1
𝑆𝑛=
2𝑛
𝑛+1.
(1)'{𝑎𝑛}%()2v
(2)Á𝑏𝑛=
(𝑛−1)(𝑛+2)
𝑎𝑛𝑎𝑛+1
W'
1{𝑏𝑛}%»𝑛)K𝑇𝑛.
è2 12-3Y2024·Ÿĥƹƈ·āĂZo7)
1{𝑎𝑛}qWÀ‡𝑎1 = 1W¾
𝑛𝑎𝑛
𝑎𝑛+1−
(𝑛+1)𝑎𝑛+1
𝑎𝑛
= 1‰
(1)'
1{𝑎𝑛}%()2v
(2)'ŕ
2 ≤(𝑎𝑛+ 1)𝑛< 3‰
k9ŧ
1‰Y2024·Ʒźƺœ·úĂZ𝑛∈𝑁*W
1 1W−3W7W−15W31W ⋅⋅⋅%k“()2×Y    Z
A‰𝑎𝑛= (2𝑛−1)cos𝑛π
B‰𝑎𝑛= (1−2𝑛)sin
𝑛π
2
C‰𝑎𝑛= 2𝑛−1
D‰𝑎𝑛= (−1)𝑛(1−2𝑛)
2‰Y2024·ŗƻƼ·ĂĬĭĮZÁ𝑎𝑛= 𝑎𝑛−1 +𝑛−1W𝑎1 = 1ģ𝑎10 = Y    Z
A‰55
B‰56
C‰45
D‰46
3‰Y23-24 úō·ŒœœƯ·ĨĩĪīZÀ‡
1{𝑎𝑛}%)řŚ𝑎𝑛+1 =
𝑛
𝑛+2𝑎𝑛WÈ𝑎1 = 1Wģ𝑎𝑛ƽY    Z
      
      
A‰
2
(𝑛+1)2
B‰
2
𝑛(𝑛+1)
C‰
1
2𝑛−1
D‰
1
2𝑛−1
4‰Y23-24 ú·ST·ûüýþZo
1{𝑎𝑛}qW𝑎1 = 1W¾𝑎𝑛+1 = 2𝑎𝑛+1Wģ{𝑎𝑛}%()×Y    Z
A‰𝑎𝑛= 2𝑛−1
B‰𝑎𝑛= 2𝑛
C‰𝑎𝑛= 2𝑛+1
D‰𝑎𝑛= 2𝑛+1
5‰Y2024·ƾƘƿǀ·kĂZÀ‡𝑇𝑛×7)
1{𝑎𝑛}%»𝑛)%,ƒW¾𝑎1 = 2,𝑇2
𝑛= 𝑎𝑛+1
𝑛
Wģ𝑎5 = Y    Z
A‰16
B‰32
C‰64
D‰128
6‰Y2024·ŷŸ·ĂĬĭĮZÀ‡
1{𝑎𝑛}řŚ(2𝑛−3)𝑎𝑛−(2𝑛−1)𝑎𝑛−1 = 4𝑛2−8𝑛+ 3(𝑛≥2,𝑛∈𝑁*),𝑎1 = 1W
ģ𝑎𝑛= Y    Z
A‰2𝑛−2
B‰2𝑛2−𝑛
C‰2𝑛−1
D‰(2𝑛−1)2
7‰Y2024·ǁœ·ĂĬĭĮZÀ‡/3
1{𝑎𝑛}%3t× 1WÁ𝑎1 = 2W¾3𝑎1,𝑎2,−𝑎3ç/0
1Wģ𝑎𝑛=
Y    Z
A‰2 × 3𝑛−1
B‰3𝑛
C‰2 × (−3)𝑛−1
D‰(−3)𝑛
8‰Y23-24 úō·ĤƘǂǃ·ĨĩĪīZÀ‡
1{𝑎𝑛}řŚ𝑎1 =
1
2W𝑎𝑛+1 = 𝑎𝑛+
1
𝑛2+𝑛Wģ{𝑎𝑛}%()×Y    Z
A‰𝑎𝑛=
1
𝑛+1W𝑛≥1W𝑛∈N∗
B‰𝑎𝑛=
3
2 +
1
𝑛W𝑛≥1W𝑛∈N∗
C‰𝑎𝑛= −
3
2−
1
𝑛W𝑛≥1W𝑛∈N∗
D‰𝑎𝑛=
3
2−
1
𝑛W𝑛≥1W𝑛∈N∗
ú9
9‰Y23-24 ú·ST·ûüýþZÀ‡
1{𝑎𝑛}řŚ𝑎1 = 1W𝑎𝑛= 𝑎1 +
1
2𝑎2 +
1
3𝑎3 +⋯+
1
𝑛−1𝑎𝑛−1(𝑛> 1)Wģ
Y    Z
A‰𝑎2 = 1
B‰
𝑎𝑛
𝑎𝑛−1 =
𝑛
𝑛−1
C‰𝑎𝑛=
𝑛
2
D‰𝑎𝑛= 1,𝑛= 1
𝑛
2 ,𝑛≥2
 
10‰Y2024·ST·ĂĬĭĮZ
1{𝑎𝑛}qWÁ𝑇𝑛= 𝑎1𝑎2⋯𝑎𝑛ƓoŜuŝWģ
1{𝑎𝑛}%()’h[Y    Z
A‰𝑎𝑛=
𝑛+1
𝑛
B‰𝑎𝑛= 2𝑛−7
C‰𝑎𝑛= −1
2
𝑛
D‰𝑎𝑛= sin
𝑛π
4
11‰Y2024·ĤƘƜƝ·ĂĬĭĮZÀ‡
1{𝑎𝑛}řŚ𝑎1 = 1W𝑎𝑛+1 =
𝑎𝑛
1+ 𝑎𝑛W𝑛∈𝑁∗Wģō1Ð=ƟƠ%[
Y   Z
A‰𝑎3 = 2−2
2
B‰Ɠo𝑛∈𝑁∗WƏÓ
1
𝑎𝑛+1 >
1
𝑎𝑛+
1
2
      
      
C‰𝑎𝑛+1 ≤
𝑛+1
𝑛+3𝑎𝑛
D‰𝑎22 >
1
92
ā9|}
12‰Y2024·ůĥƊƥ·ĂĬĭĮZÀ‡k
1
0,2,−6,12,−20,30,⋯WģÃ
1%()’h•–×      
.
13‰Y2023·ST·ĂĬĭĮZÀ‡
1{𝑎𝑛}řŚ𝑎1 = 1W𝑎2 = 2W𝑎𝑛+1 + 𝑎𝑛−1 +2𝑎𝑛= 0 (𝑛≥2,𝑛∈N∗)ģÃ

1%()2×      
‰
14‰Y2024·ƾĥœƋ·kĂZÀ‡
1{𝑎𝑛}řŚ𝑛𝑎𝑛+1−(𝑛+ 1)𝑎𝑛= 2W𝑎1 = 1Wģ
1{𝑎𝑛}%()2×      
‰
Ū9Ji
15‰Y23-24 ú«·ST·ûüýþZo
1{𝑎𝑛}qW𝑎1 = 2W𝑎17 = 66W()2𝑎𝑛= 𝑝𝑛+ 𝑞WĢq pWq
×P
W𝑝≠0‰
(1)'{𝑎𝑛}%()2v
(2)88 [ţ[
1{𝑎𝑛}q%)Ɣ
16‰Y2024·ST·ĂĬĭĮZÀ‡
1{𝑎𝑛}řŚ𝑎1 +2𝑎2 +3𝑎3 +⋯+ 𝑛𝑎𝑛= (𝑛−1)2𝑛+1‰
(1)'
1{𝑎𝑛}%()2v
(2)Á𝑏𝑛=
𝑛𝑎𝑛+1
𝑛2+3𝑛+2W'
1{𝑏𝑛}%»𝑛)K𝑆𝑛‰
17‰Y2024·ůĥDŽĀ·kĂZÀ‡
1{𝑎𝑛}%»𝑛)K×𝑆𝑛W¾𝑎1 = 3,𝑆𝑛+1 + 𝑆𝑛= (𝑛+ 1)𝑎𝑛+1.
(1)'{𝑎𝑛}%()2v
(2)Á𝑏𝑛=
1
𝑎𝑛𝑎𝑛+1W'
1{𝑏𝑛}%»𝑛)K𝑇𝑛.
      
      
18‰Y2024·ŒŬDždž·ĂĬĭĮZśµ)Ƨ×7
%
1{𝑎𝑛}%»𝑛)K×𝑆𝑛WÀ‡ 𝑆𝑛[
𝑎𝑛−1
2 $
𝑎𝑛+3
2 %/0
q).
(1)'{𝑎𝑛}%()2v
(2)Ɖ𝑏𝑛=
𝑎𝑛+1
𝑆𝑛𝑆𝑛+1 +
𝑎2
𝑛+1
𝑆𝑛𝑆𝑛+1W
1{𝑏𝑛}%»𝑛)K×𝑇𝑛WŕŦ
𝑇𝑛−4𝑛< 2.
19‰Y2024·Ūū^Ÿ·āĂZÀ‡/0
1{𝑎𝑛}%0× 4W¾𝑎2 +2W𝑎3W𝑎5−2ç/3
1W
1{𝑏𝑛}%»
n )K×𝑆𝑛W𝑏1 = 2¾𝑆𝑛= 2𝑆𝑛−1 +2(𝑛≥2).
(1)'
1{𝑎𝑛}9{𝑏𝑛}%()2v
(2)Ɖ𝑐𝑛= 𝑎𝑛𝑏𝑛(𝑛∈𝑁∗)W'
1{𝑐𝑛}%» n )K𝑇𝑛.