专题6.5数列求和九大题型（原卷版）_11.4-2_2026-11-04

      
      
专题6.5 数列求和【九大题型】

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 1  
Ò 1+2024·ÓÔÕÖ·×Ø-MN
B{𝑎𝑛}3P𝑛'%𝑆𝑛,𝑎1 = 8YÙ𝑆9 = 0.
(1)$𝑆𝑛Ú
(2)¤{𝑏𝑛}MO
BY𝑏1 =
𝑆4
5 ,𝑏2 = −𝑎6Y${𝑏𝑛}¿'.
Û 1-1+2024·ÓÔ©Ü·ØÝÞß-àœ{𝑎𝑛}]MN
BY𝑎1 = 1YÙ𝑎1Y𝑎3Y𝑎9©MO
Bž
(1)$
B{𝑎𝑛}3NÚ
(2)$
B{2𝑎𝑛}3P𝑛'%𝑆𝑛ž
Û 1-2+2024·áâ·lØ-àœ𝑆𝑛
B{𝑎𝑛}3P n '%Yãä𝑆𝑛=
1
2𝑎2
𝑛+
1
2𝑎𝑛−1(𝑛∈N∗)YÙ𝑎1,𝑎2,𝑎3
,𝑎4,𝑎5©MO
BYå𝑛≥5‰Y𝑎𝑛> 0ž
(1)$æ
å𝑛≥5‰Y{𝑎𝑛}©MN
BÚ
(2)${𝑎𝑛}3P n '%𝑆𝑛ž
Û 1-3+2024·çèéÖ·×Ø-àœ
B{𝑎𝑛}ãä𝑎1 =
1
4Y𝑎𝑛Y
3
2𝑎𝑛+1Y2𝑎𝑛𝑎𝑛+1©MN
B.
(1)$æ

B
1
𝑎𝑛−1 ]MO
BY,$˜{𝑎𝑛}3¿'Ú
(2)ê{𝑎𝑛}3P n '%𝑆𝑛Yæë
3
8 1−1
3
𝑛
≤𝑆𝑛<
5
12.
      
      
 2   !"#$%
Ò 2+2024·ìí·îØ-àœMN
B{𝑎𝑛}ãä𝑎2 = 5Y𝑎3 + 𝑎4 = 22ž
(1)${𝑎𝑛}3¿'Ú
(2)$
B{𝑎𝑛⋅2𝑛}3P n '%ž
Û 2-1+2024·ïðçñòç·lØ-ó𝑛∈N∗Y¤
B{𝑎𝑛}3P𝑛'%𝑆𝑛YÙ𝑎𝑛]2𝑆𝑛3MNy'Ú
(1)$
B{𝑎𝑛}3¿'Ú
(2)¤{𝑎𝑛⋅𝑏𝑛}]s2Ë'Y4N3MN
BY$
B{𝑏𝑛}3P𝑛'%𝑇𝑛ž
Û 2-2+2024·ôèõí·ØÝÞß-àœ±'gö
3
B{𝑎𝑛}3P𝑛'%𝑆𝑛,2𝑆𝑛= 𝑛(𝑎𝑛+ 1)Ù
𝑎2 =
3
2𝑎1.
(1)${𝑎𝑛}3¿'Ú
(2)¤𝑏𝑛=
𝑎𝑛
2𝑛,$
B{𝑏𝑛}3P𝑛'%𝑇𝑛.
Û 2-3+2024·÷ø·ØÝÞß-
B{𝑎𝑛}]MN
BYÇP n '%𝑆𝑛Y
B{𝑏𝑛}]MO
BY𝑆3−𝑆2
= 3Y𝑎1 + 𝑎2 = 3Y𝑏𝑛> 0Y𝑏1 ⋅𝑏2 = 𝑏3Y2𝑏1 + 𝑏2 = 𝑏3ž
      
      
(1)$
B{𝑎𝑛}>{𝑏𝑛}3¿'Ú
(2) 𝑎𝑛
𝑏𝑛3P n '%𝑇𝑛Y$æ
1
2 ≤𝑇𝑛< 2ž
 3  &'"($%
Ò 3+2024·ôèùú·ØÝÞß-àœ
B{𝑎𝑛}ãä𝑎1 = 2,𝑎𝑛+1 = 2𝑛+1𝑎𝑛
𝑎𝑛+2𝑛.
(1)æë

B 2𝑛
𝑎𝑛]MN
BÚ
(2)ó𝑏𝑛=
𝑎𝑛+1𝑎𝑛+2
𝑎𝑛
Y${𝑏𝑛}3P n '%𝑇𝑛.
Û 3-1+2024·ïðçûüý·ØÝÞß-ê𝑆𝑛
B{𝑎𝑛}3P n '%Y
𝑆𝑛−𝑛]Ë'Ng 1 3
MN
Bž
(1)$
B{𝑎𝑛}3¿'Ú
(2)ó𝑏𝑛=
(−1)𝑛(𝑎𝑛+1)
𝑆𝑛
Y$
B{𝑏𝑛}3P 2024 '3%𝑇2024ž
Û 3-2+2024·þÿðĀ·îØ-¤
B{𝑎𝑛}]N 1 3MN
BYÙ𝑎3 = 2YC(𝑎𝑛,𝑏𝑛)|ƒ
𝑓(𝑥) = 3𝑥3āĂă(𝑛∈N∗)Yê
B{𝑏𝑛}3P𝑛'%𝑆𝑛. 
(1)$
B{𝑎𝑛},{𝑏𝑛}3¿';
(2)ó𝑐𝑛=
𝑏𝑛
4𝑆𝑛𝑆𝑛+1Yê
B{𝑐𝑛}3P𝑛'%𝑇𝑛Yæë:𝑇𝑛<
1
12. 
      
      
Û 3-3+2024·ĄíąĆ·ØÝÞß-àœMN
B{𝑎𝑛}3P n '%𝑆𝑛YÙ{𝑆𝑛+ 𝑛2}ć]MN
Bž
(1)$
B{𝑎𝑛}3NÚ
(2)¤𝑎1 = −1Y$
B
1
𝑎𝑛𝑎𝑛+1 3P n '%𝑇𝑛ž
 4  )*+,'-$%
Ò 4+2024·Ĉç·ØÝÞß-àœ
B{𝑎𝑛}N‚ĉ3MN
BYÇP n '%𝑆𝑛Y𝑆7 = 49YÙ
𝑎2Y𝑎5Y𝑎14©MO
Bž
(1)${𝑎𝑛}3¿'Ú
(2)¤
B{𝑎𝑛+ 𝑏𝑛}]O 3 3MO
BYÙ𝑏3 = 22Y${𝑏𝑛}3P n '%𝑇𝑛ž
Û 4-1+2024·ýè·îØ-àœMN
B{𝑎𝑛}3N𝑑> 0YP𝑛'%𝑆𝑛YÙ𝑎3𝑎6 = −5Y𝑆8 = −16.
(1)$
B{𝑎𝑛}3¿'Ú
(2)¤𝑏𝑛= 𝑎𝑛,𝑛= 2𝑘−1
2𝑛,𝑛= 2𝑘(𝑘∈𝑁∗)
 
Y$
B{𝑏𝑛}3P2𝑛'%𝑇2𝑛.
Û 4-2+2024·ïðç·îØ-àœMN
B{𝑎𝑛}3N𝑑> 0Y𝑎2𝑎83MNy' 5YÙ𝑎4𝑎6 = 24.
(1)$
B{𝑎𝑛}3¿'Ú
      
      
(2)ó𝑏𝑛=
𝑎𝑛,𝑛为奇数，
1
𝑎𝑛𝑎𝑛+2 ,𝑛为偶数，
 
$
B{𝑏𝑛}3P 20 '%𝑇20.
Û 4-3+2024·ĄíĊĆ·îØ-àœMN
B{𝑎𝑛}ãä
𝑎1 = 2YÙ𝑎1Y𝑎2Y𝑎4©MO
Bž
(1)$
B{𝑎𝑛}3¿'Ú
(2)¤MN
B{𝑎𝑛}3N‚ĉÙ
B{𝑏𝑛}ãä
𝑏𝑛=
4𝑛2
(𝑎𝑛−1)(𝑎𝑛+1)Y$
B{𝑏𝑛}3P𝑛'%𝑇𝑛ž
 5  ./"0$%
Ò 5+2024·ăċ·ØÝÞß-àœ𝑓(𝑥) =
1
2𝑥2 +
1
2𝑥Y
B{𝑎𝑛}3P𝑛'%𝑆𝑛YC(𝑛,𝑆𝑛)(𝑛∈N*)g|ƒ
𝑦= 𝑓(𝑥)3āĂă.
(1)$
B{𝑎𝑛}3¿'Ú
(2)¤𝑔(𝑥) =
4𝑥
4𝑥+2YČ𝑏𝑛= 𝑔
𝑎𝑛
2025 (𝑛∈N*)Y$
B{𝑏𝑛}3P 2024 '%𝑇2024.
Û 5-1+23-24 ×A·ÓÔ©Ü·čĎJď-àœ
B{𝑎𝑛}ãä
𝑎1
2 +
𝑎2
22 +
𝑎3
23 + ⋅⋅⋅+
𝑎𝑛
2𝑛= 𝑛(𝑛∈𝑁∗)Y
B{𝑏𝑛}ãä𝑏𝑛=
1
𝑎𝑛+250.
(1)$
B{𝑎𝑛}3¿'Ú
(2)$𝑏𝑛+ 𝑏100−𝑛3…Ú
(3)$𝑏1 + 𝑏2 + 𝑏3 + ⋅⋅⋅+ 𝑏993….
      
      
Û 5-2+23-24 ×A·ĄíĐĆ·čĎJď-àœ
B{𝑎𝑛}ãä
𝑎1
2 +
𝑎2
22 +⋯+
𝑎𝑛
2𝑛= 𝑛(𝑛∈𝑁∗)Y
B{𝑏𝑛}
ãä𝑏𝑛=
1
𝑎𝑛+250ž
(1)$
B{𝑎𝑛}3¿'Ú
(2)$
B
𝑛
𝑎𝑛3P n '%𝑆𝑛Ú
(3)$
B{𝑏𝑛}3P 99 '3%𝑇993…ž
Û 5-3+23-24 ×A·Z[·đPÞď-àœƒ
𝑓(𝑥) =
1
4𝑥+2(𝑥∈𝑅).
(1)$æ𝑓(𝑥) +𝑓(1−𝑥)<…Ú
(2)¤
B{𝑎𝑛}3¿'𝑎𝑛= 𝑓
𝑛
𝑚+𝑚öĒ
Y𝑛= 1Y2Y⋯Y𝑚-Y$
B{𝑎𝑛}3P𝑚'%𝑆𝑚Ú
 6  12+-1-n 34$%
Ò 6+2024·áâēĆ·ØÝÞß-àœ
B{𝑎𝑛}ãä𝑎1 = 1Y𝑎𝑛> 0Y𝑆𝑛]
B{𝑎𝑛}3P𝑛'%Y²Ĕĕ
𝑛∈N ∗Y22𝑆𝑛= 2𝑎2
𝑛+ 𝑎𝑛−1
(1)$
B{𝑎𝑛}3¿'Ú
(2)ó𝑏𝑛= (−1)𝑛−1𝑎𝑛Y${𝑏𝑛}3P 100 '3%.
      
      
Û 6-1+23-24 ×A·ĖėĘý·ęy-ó{𝑎𝑛}]MN
BY{𝑏𝑛}]Oei 0 3MO
BYàœ𝑎1 =
𝑏1 = 2Y𝑏2 = 𝑎2Y𝑏3 = 𝑎2 +4ž
(1)${𝑎𝑛}%{𝑏𝑛}3¿'Ú
(2)ó𝑐𝑛= (−1)𝑛𝑎𝑛+ 𝑏𝑛Y$
B{𝑐𝑛}3P2𝑛'%𝑇2𝑛ž
Û 6-2+2024·ôèõí·×Ø-àœMO
B{𝑎𝑛}3±'gö
YP n '%𝑆𝑛YÙãä𝑎1 + 𝑎2
= 3Y𝑆4 = 15.
(1)$
B{𝑎𝑛}3¿'Ú
(2)¤
B{𝑏𝑛}ãä𝑏𝑛= 𝑎𝑛+ (−1)𝑛(3𝑛−1)Y$
B{𝑏𝑛}3P 2n '%𝑇2𝑛.
Û 6-3+2024·ôèùú·ØÝÞß-ê𝑆𝑛
B{𝑎𝑛}3P𝑛'%Yàœ𝑎1 = 1,𝑛𝑆𝑛+1−(𝑛+ 1)𝑆𝑛= 𝑛2
+𝑛.
(1)${𝑎𝑛}3¿'Ú
(2)¤𝑏𝑛= (−1)𝑛𝑎𝑛+ [(−1)𝑛+ 1]2𝑛Y$
B{𝑏𝑛}3P2𝑛'%𝑇2𝑛.
 7  56'7$%
Ò 7+2024·ýė·×Ø-àœ{𝑎𝑛}]N‚ 0 3MN
BYÇP 4 '% 16YÙ𝑎1,𝑎2,𝑎5©MO
B.
(1)$
B{𝑎𝑛}3¿'Ú
(2)ó𝑏𝑛=
2𝑎𝑛,𝑛为奇数
1
𝑎𝑛𝑎𝑛+2 ,𝑛为偶数
 
Y$
B{𝑏𝑛}3P2𝑛'%𝑇2𝑛.
      
      
Û 7-1+2024·ôèùú·ØÝÞß-ê
B{𝑎𝑛}3P𝑛'%𝑆𝑛Yàœ𝑎𝑛≥1Ù2𝑆𝑛= 𝑎2
𝑛+𝑛ž
(1)æë
{𝑎𝑛}]MN
BÚ
(2)ê𝑏𝑛= 2𝑎𝑛,𝑛为奇数
𝑎𝑛,𝑛为偶数
 
Y$
B{𝑏𝑛}3P 2n '%𝑇2𝑛ž
Û 7-2+2024·þÿĚÖ·×Ø-àœ
B{𝑎𝑛}%{𝑏𝑛}3±'göYÙ𝑎3 = 18𝑏1Y{𝑏𝑛}]O 3 3MO

Bž
B{𝑎𝑛}3P n '%𝑆𝑛ãä4𝑆𝑛= 𝑎2
𝑛+2𝑎𝑛ž
(1)$
B{𝑎𝑛}Y{𝑏𝑛}3¿'Ú
(2)ó𝑐𝑛=
𝑏𝑛+3
(𝑏𝑛+3−3)(𝑏𝑛+3−1) + 𝑎𝑛cos𝑛πY$
B{𝑐𝑛}3P n '%𝑇𝑛ž
Û 7-3+2024·þÿěĜ·îØ-ó𝑆𝑛
B{𝑎𝑛}3P𝑛'%Yàœ𝑎1 = 1,𝑆4 = 10YÙ 𝑆𝑛
𝑛MN
B.
(1)${𝑎𝑛}3¿'Ú
(2)¤𝑏𝑛=
𝑎𝑛,𝑛为奇数
1
𝑎𝑛𝑎𝑛+2 ,𝑛为偶数
 
Y${𝑏𝑛}3P2𝑛'%𝑇2𝑛.
      
      
 8  89:;&'$%
Ò 8+2024·þÿěĜ·×Ø-àœ
B{𝑎𝑛}ãä𝑎1 = 2Y𝑎𝑛+1 =
2𝑎𝑛−1
𝑎𝑛.
+1-æë

B
1
𝑎𝑛−1 ]MN
BÚ
+2-Č𝑏𝑛=
1
𝑎1𝑎2⋯𝑎𝑛Yæë
𝑏12 + 𝑏2
2 +⋯+ 𝑏2
𝑛< 1.
Û 8-1+23-24 îă·ôèĝĆ·čĎJď-àœ
B{𝑎𝑛}3P n '%𝑆𝑛Y𝑎1 = 1Y𝑎𝑛+1 =
𝑎𝑛
𝑎𝑛+4ž
+1-$æ
1
𝑎𝑛+ 1
3 MO
BÚ
+2-$æ
𝑆𝑛<
3
2ž
Û 8-2+23-24 lA·ÓÔĞý·ęÌ-àœ
B{𝑎𝑛}ãä𝑎1 = 1Y2𝑎𝑛+1𝑎𝑛+ 𝑎𝑛+1−𝑎𝑛= 0YČ𝑏𝑛=
1
𝑎𝑛Yó
B{𝑏𝑛}P n '%𝑆𝑛ž
(1)$æ

B{𝑏𝑛}MN
BÚ
(2)¤ğ|𝑛∈𝑁+YĠ‚M𝑎1𝑎2 + 𝑎2𝑎3 + ⋅⋅⋅+ 𝑎𝑛𝑎𝑛+1 ≥(𝑛+ 18)𝜆©ġY$Ģ
𝜆3ģ…ĤĥÚ
(3)óö'
B{𝑐𝑛}ãä𝑐𝑛2 = 1 +
2
𝑆𝑛+1Y$æ
𝑐1 + 𝑐2 + ⋅⋅⋅+ 𝑐𝑛< 𝑛+ 1−
1
𝑛+1ž
Û 8-3+2024·ĖėĦÖ·lØ-ħ
YĨ­ĩ
žĪ3<=®A
¤Ē
𝑎īsĒ
𝑚(𝑚≠0)īÆ3
Ĭöĭ]Ē
ÅĮ2į
Yİı»­𝑎𝑚3IJ
Y­𝑚𝑎3ħ
žóöĒ
𝑎ij2𝑘¥öħ
Yê𝑎1Y
      
      
𝑎2Y…Y𝑎𝑘−1Y𝑎𝑘+𝑎1 < 𝑎2 <⋅⋅⋅< 𝑎𝑘-ž
(1)å𝑘= 4‰Y¤öĒ
𝑎3𝑘¥öħ
µ©MO
BYĴĵ˜l¥𝑎3…Ú
(2)å𝑘≥4‰Y¤𝑎2−𝑎1Y𝑎3−𝑎2Y…Y𝑎𝑘−𝑎𝑘−1µ©MO
BY$æ
𝑎= 𝑎𝑘−1
2
(𝑘≥4)Ú
(3)ê𝐴= 𝑎1𝑎2 + 𝑎2𝑎3 +⋯+ 𝑎𝑘−1𝑎𝑘Y$æ
𝐴< 𝑎2ž
 9  <=>?@A3
B$%
Ò 9+2024·ôè·îØ-
B{𝑎𝑛}3P𝑛'3„e…ê𝑀𝑛Y¹𝑀𝑛= max{𝑎1,𝑎2, ⋅⋅⋅,𝑎𝑛}ÚP𝑛'3„f…
ê𝑚𝑛Y¹𝑚𝑛= min{𝑎1,𝑎2, ⋅⋅⋅,𝑎𝑛}YČ𝑝𝑛= 𝑀𝑛−𝑚𝑛Y,Ķ
B{𝑝𝑛}­{𝑎𝑛}3“ķ©
B”ž
(1)ó
B{𝑝𝑛}3“ķ©
B”{𝑞𝑛}Y$æ
𝑝𝑛= 𝑞𝑛Ú
(2)¤𝑎𝑛= 2𝑛−3𝑛Y$Çķ©
B{𝑝𝑛}3P𝑛'%ž
Û 9-1+2024·‘ĸ·ØÝÞß-²i
B{𝑎𝑛}Y<=Δ𝑎𝑛= 𝑎𝑛+1−𝑎𝑛(𝑛∈𝑁∗)Yãä𝑎1 = 𝑎2 = 1,Δ(Δ𝑎𝑛)
= 𝑚(𝑚∈𝑅)Yê𝑓(𝑚,𝑛) = 𝑎1𝑚+ 𝑎2𝑚2 +⋯+ 𝑎𝑛𝑚𝑛Y­𝑓(𝑚,𝑛)¦
B{𝑎𝑛}ķ©3“𝑚−ƒ
”ž
(1)Ĺĵ˜“2−ƒ
” 𝑓(2,𝑛)Y,$𝑓(2,3)3…Ú
(2)¤“1−ƒ
” 𝑓(1,𝑛) ≤15Y$ n 3„e…Ú
(3)êƒ
𝑆(𝑥) = 𝑥+ 2𝑥2 +⋯+ 𝑛𝑥𝑛Yǽƒ
𝑆′(𝑥)Yæë
“𝑚−ƒ
” 𝑓(𝑚,𝑛) = 𝑚2
2 𝑆′(𝑚)−
3𝑚
2
𝑆(𝑚) + (𝑚+ 1)
𝑛
𝑖=1𝑚′ž
Û 9-2+2024·ýėĺù·ØÝÞß-àœ
B{𝑎𝑛}]Ļļ¶Ľ
BYÇ
…
1,1,2,3,5,8,13, 
      
      
21,34 ⋅⋅⋅⋅⋅⋅.¸l
Bs®Aľ¼3U<=
𝑎1 = 1，𝑎2 = 1，𝑎𝑛+2 = 𝑎𝑛+1 + 𝑎𝑛 (𝑛∈𝑁*).
B{𝑏𝑛}²iĿ<
3öĒ
𝑘Y¤ğ|öĒ
𝑛ĠÆ𝑏𝑘+𝑛= 𝑏𝑘+ 𝑏𝑛©ġYª­
B{𝑏𝑛}“𝑘č¨)À
B”.
(1)àœ
B{𝑐𝑛}ãä𝑐𝑛= 𝑚𝑎𝑛 （𝑛∈𝑁*，𝑚∈𝑅）.ŀŁ]ł²∀𝑚∈𝑅YÑğ|Ŀ<3öĒ
𝑘YĠÆ
B{𝑐𝑛}
“𝑘č¨)À
B”Y,Ńë¦.
(2)ó
B{𝑑𝑛}3P𝑛'%𝑆𝑛= 3𝑛−𝑎 (𝑎≥0)Y
+i-¤
B{𝑑𝑛}“1č¨)À
B”Y$˜ń‡Ņņ3Ģ
𝑎3…Ú
+ii-|+i-73PŇAY¤
B{𝑓𝑛}ãä𝑓𝑛=
𝑎𝑛
𝑆𝑛Y𝑛∈𝑁*YÇP𝑛'%𝑇𝑛.æë
å𝑛∈𝑁*Ù𝑛≥3‰Y𝑇𝑛<
𝑎2
1 + 𝑎2
2 + 𝑎2
3 + ⋅⋅⋅⋅⋅⋅+ 𝑎2
𝑛−𝑎𝑛𝑎𝑛+1 +1©ġ.
Û 9-3+2024·çè·ØÝÞß-İ[ňʼn
ŊŋŌō|Ŏ3ŏÓňŐőŒlœy²čMN
B$
%2Ŕŕ3ŖŗY¹“Ř´ř”ž²i
B𝑎1,𝑎2, ⋅⋅⋅,𝑎𝑛, ⋅⋅⋅Y①YmX×'ŚYśl'ĪPŜ"ŝl'3N
µ©
B𝑎11,𝑎12, ⋅⋅⋅,𝑎1(𝑛−1), ⋅⋅⋅Y②Y­Ş
B②
B①3lčN)
BYÇy𝑎1𝑖= 𝑎𝑖+1−𝑎𝑖
(𝑖= 1,2, ⋅⋅⋅,𝑛−1, ⋅⋅⋅)Ú²i
B②YmX×'ŚYśl'ĪPŜ"ŝl'3Nµ©
B𝑎21,𝑎22, ⋅⋅⋅,𝑎2(𝑛−2)
, ⋅⋅⋅Y③Y­Ş
B③
B①3×čN)
BYÇy𝑎2𝑖= 𝑎1(𝑖+1)−𝑎1𝑖(𝑖= 1,2, ⋅⋅⋅,𝑛−2, ⋅⋅⋅) ⋅⋅⋅⋅⋅⋅şŠăš
ŢYX𝑟ţÆŤ
B𝑎𝑟1,𝑎𝑟2, ⋅⋅⋅,𝑎𝑟(𝑛−𝑟), ⋅⋅⋅Y④Yª­
B④
B①3𝑟čN)
BYÇy𝑎𝑟𝑖= 𝑎(𝑟−1)(𝑖+1)
−𝑎(𝑟−1)𝑖(𝑖= 1,2, ⋅⋅⋅,𝑛−𝑟, ⋅⋅⋅)Y¤
B{𝑎𝑛}3𝑟(𝑟≥2)čN)
B]QĉT
BYª­
B{𝑎𝑛}𝑟čMN
B
+zčMN
B-ž
(1)¤čMN
B{𝑎𝑛}3,4,9,18,31,48, ⋅⋅⋅Y$
B{𝑎𝑛}3¿'Ú
(2)¤𝑟čMN
B{𝑏𝑛}3¿'𝑏𝑛= (2𝑛−1)4ž
+ⅰ-$𝑟3…Ú
+ⅱ-$
B{𝑏𝑛}3P𝑛'%𝑆𝑛ž
ť
12 + 22 + ⋅⋅⋅+ 𝑛2 =
𝑛(𝑛+1)(2𝑛+1)
6
ž
      
      
l>Ŧ
1ž+2024·ŧ·×Ø-àœMN
B{𝑎𝑛}3P𝑛'%𝑆𝑛Y¤
𝑎7
𝑎8 = −1Yª𝑆10 = +    -
Až𝑆4
Bž𝑆5
Cž𝑆6
Dž𝑆7
2ž+2024·ÓÔ_ç·ØÝÞß-|
B{𝑎𝑛}yYàœ𝑎1 =
1
2Y(𝑛+ 2)𝑎𝑛+1 = 𝑛𝑎𝑛YªĪ3P 30 '3%
+   -
Až
19
29
Bž
28
29
Cž
29
30
Dž
30
31
3ž+2024·ĄŨ·ØÝÞß-àœ{𝑎𝑛}]±'gö
3MO
BY𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 = 10Y𝑎1𝑎2𝑎3
𝑎4𝑎5𝑎6 = 8Yª
1
𝑎1 +
1
𝑎2 +
1
𝑎3 +
1
𝑎4 +
1
𝑎5 +
1
𝑎6 = +    -
Až2
Bž3
Cž4
Dž5
4ž+23-24 îă·ũíŪū·čĎJď-àœ
B{𝑏𝑛}]O q+𝑞≠1-3ö'MO
BYÙ2ln𝑏1012
= 0Y¤𝑓(𝑥) =
4
1+𝑥2Yª𝑓(𝑏1) +𝑓(𝑏2) +⋯+ 𝑓(𝑏2023) = +    -
Až4069
Bž2023
Cž2024
Dž4046
5ž+2024·ìŨŬŊŭ·îØ-àœ
B{𝑎𝑛}3P n '%𝑆𝑛YÙãä𝑎1 = 1,𝑎𝑛+1 = 𝑎𝑛+ 1,𝑛为奇数
2𝑎𝑛,𝑛为偶数
 
Yª
𝑆100 = +    -
Až3 × 251−156
Bž3 × 251−103
Cž3 × 250−156
Dž3 × 250−103
6ž+2024·ÓÔŮůŰ·îØ-
B{𝑎𝑛}3P𝑛'%𝑆𝑛Y𝑎1 = −1Y𝑛𝑎𝑛= 𝑆𝑛+𝑛(𝑛−1)(𝑛∈N∗)Yó𝑏𝑛= (−1)𝑛
𝑎𝑛Yª
B{𝑏𝑛}3P 51 'k%+    -
Až−149
Bž−49
Cž49
Dž149
7ž+2024·÷øŨű·ØÝÞß-ó
B{𝑎𝑛}ãä𝑎1 +2𝑎2 +3𝑎3 + ⋅⋅⋅+𝑛𝑎𝑛= 2𝑛+ 1(𝑛∈𝑁*)Yª
B
𝑎𝑛
𝑛+1 3
P 5 '%+   -
Až
5
3
Bž
8
5
Cž
12
7
Dž
13
6
8ž+2024·ÓÔŲĆ·ØÝÞß-àœ
B{𝑎𝑛}3±'gö
Y𝑎1 = 1,𝑎𝑛+1−𝑎𝑛=
1
𝑎𝑛+1+𝑎𝑛Y¤[𝑥]ųŴ‚ŵŶ
𝑥3„eĒ
Yª[𝑎1] + [𝑎2] +⋯+ [𝑎100] = +   -
      
      
Až615
Bž620
Cž625
Dž630
×>c
9ž+2023·ýėŷŠ·ØÝÞß-àœ
B{𝑎𝑛}yY𝑎𝑛= 2𝑛+ 1,𝑏𝑛= 2𝑛ª+    -
Až{𝑎𝑛𝑏𝑛}3P 10 '%19 × 210 +2
Bž{(−1)𝑛𝑎𝑛}3P 100 '% 100
Cž
1
𝑎𝑛𝑎𝑛+1 3P𝑛'%𝑇𝑛,𝑇𝑛<
1
6
Dž 𝑎𝑛+ 64
𝑎𝑛3„f'16
1
7
10ž+2024·ŸŹ·ØÝÞß-àœ|N‚ 0 3MN
B{𝑎𝑛}yY𝑎4 = −5,𝑎5]𝑎2𝑎63MOy'Y
B{𝑏𝑛}
3P𝑛'%𝑆𝑛YÙ𝑏𝑛=
1
𝑎𝑛𝑎𝑛+1Yª+    -
Až𝑎𝑛= 2𝑛−13
Bž∀𝑛∈N*,𝑏𝑛≥−1
Cž𝑆𝑛= −
1
11−
1
2𝑛−11
Dž∀𝑛∈N*,𝑆6 ≤𝑆𝑛≤𝑆5
11ž+2024·źÖŻż·îØ-àœMN
B{𝑎𝑛}3P n '%𝑆𝑛YÙ𝑆4 = 4𝑆2,𝑎2𝑛= 2𝑎𝑛+1(𝑛∈N∗)Yª
+    -
Až𝑎𝑛= 2𝑛−1
Bž𝑆𝑛= 𝑛2
Cž
B
1
𝑎𝑛𝑎𝑛+1 3P n '%
2𝑛
2𝑛+1
Dž
B{𝑎𝑛+ 2𝑛}3P n '%2𝑛+1 + 𝑛2−2
î>Žž
12ž+2024·ýėſƀ·îØ-àœMN
B{𝑎𝑛}3N𝑑≠0YË' 𝑎1 =
1
2 Y𝑎4]𝑎2𝑎83MOy'Yê 𝑆𝑛 

B{𝑎𝑛}3P𝑛'%Yª𝑆20 =      
.
13ž+2024·ÓÔ·îØ-|
B{𝑎𝑛}yYàœ𝑎1 =
1
2Y(𝑛+ 2)𝑎𝑛+1 = 𝑛𝑎𝑛Yª
B{𝑎𝑛}3P 2024 '%𝑆2024 =
      
ž
14ž+2024·çèƁƂ·ØÝÞß-àœ
B{𝑎𝑛}]MN
BY𝑏𝑛=
𝑎𝑛−8,𝑛为奇数
2𝑎𝑛+ 1,𝑛为偶数
 
Yê𝑆𝑛Y𝑇𝑛)«{𝑎𝑛}Y
{𝑏𝑛}3P𝑛'%Y¤𝑆3 = 18Y𝑇2 = 10Yª𝑇20 =      
ž
Ó>tu
15ž+2024·_ƃƄƅƆ·îØ-àœ
B{𝑎𝑛}3P n '%𝑆𝑛Y𝑎1 = 3Y𝑆𝑛= 1 + 𝑎𝑛+1ž
(1)æë

B{𝑆𝑛−1}]MO
BY,$𝑆𝑛Ú
(2)$
B
1
𝑎𝑛3P n '%𝑇𝑛ž
      
      
16ž+2024·ÓÔ_ç·îØ-àœMN
B{𝑎𝑛}3N 4YÙ𝑎2 +2Y𝑎3Y𝑎5−2©MO
BY
B{𝑏𝑛}3P
n '%𝑆𝑛Y𝑏1 = 2Ù𝑆𝑛= 2𝑆𝑛−1 +2(𝑛≥2).
(1)$
B{𝑎𝑛}>{𝑏𝑛}3¿'Ú
(2)ó𝑐𝑛= 𝑎𝑛𝑏𝑛(𝑛∈𝑁∗)Y$
B{𝑐𝑛}3P n '%𝑇𝑛.
17ž+2024·ôèùú·ØÝÞß-àœMN
B{𝑎𝑛}3P𝑛'%𝑆𝑛YÙ𝑆5 = 45,𝑎4 + 𝑎6 = 26.
(1)$
B{𝑎𝑛}3¿'Ú
(2)¤𝑏𝑛=
1
𝑆𝑛−3𝑛Y$
B{𝑏𝑛}3P 10 '%.
18ž+2024·ýėƇƈ·×Ø-àœ
B{𝑎𝑛},{𝑏𝑛}ãä𝑎2𝑛−1 = 𝑏2𝑛−1 +12𝑚,𝑎2𝑛= 𝑚𝑏2𝑛,𝑚T
Y¤{𝑎𝑛}M
N
BYÙ𝑏4−𝑏2 = 2(𝑏3−𝑏1) = 2(𝑎1 + 𝑏1) = 8ž
(1)$𝑚3…Ž{𝑎𝑛}3¿'Ú
(2)${𝑏𝑛}3P2𝑛'%𝑆2𝑛ž
19ž+2024·÷øìŨ·×Ø-àœ{𝑎𝑛}]MN
BYÇPn'%𝑆𝑛，{𝑏𝑛}]MO
BYàœ𝑎1 = 1，𝑆3 = 6Y
𝑏1 = 𝑎2，𝑎8]𝑎4%𝑏43MOy'.
      
      
(1)${𝑎𝑛}%{𝑏𝑛}3¿'Ú
(2)$
B 𝑎𝑛
𝑏𝑛3P𝑛'%𝑇𝑛Ú
(3)ê𝑐𝑛=
𝑏𝑛−1
𝑏𝑛+1−1Y$æ
𝑛
2−
1
2 +
1
2𝑛+1 <
𝑛
𝑖=1𝑐𝑖<
𝑛
2−
1
4 +
1
2𝑛+2.