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222-9+11+12:2*14+14 = ? ( )

Toán Lớp 5: Cho A = 1/2×2 + 1/3×3 + 1/4×4 +-.+1/2021×2021 a) CMR : A < 1 b)A <3/4. c) CMR : A <13/18

Home/ Toán Lớp 5: Cho A = 1/2×2 + 1/3×3 + 1/4×4 +-.+1/2021×2021 a) CMR : A < 1 b)A <3/4. c) CMR : A <13/18

Toán Lớp 5: Cho A = 1/2×2 + 1/3×3 + 1/4×4 +-.+1/2021×2021 a) CMR : A < 1 b)A <3/4. c) CMR : A <13/18

Tháng Một 20, 2023 Toán học 2 Comments 0 views

Toán Lớp 5: Cho A = 1/2×2 + 1/3×3 + 1/4×4 +…..+1/2021×2021 a) CMR : A < 1 b)A <3/4. c) CMR : A <13/18

Comments ( 2 )

  1. ngocbichchau
    ngocbichchau
    20 Tháng Một, 2023 at 9:36 sáng
    Reply
    a) Ta thấy : 1/2.2<1/1.2 
                      1/3.3<1/2.3  
              ……………………………….
                          1/2021.2021<1/2020.2021 
    ⇒ 1/2.2+1/3.3+…..+1/2021.2021<1/1.2+1/2.3+….+1/2020.2021 
                                                     A<1-1/2+1/2-1/3+….+1/2020-1/2021<1 
    ⇒A <1 (đccm) 
     b) ta thấy 1/2.2=1/2.2
                   1/3.3<1/2.3
                    ………
                1/2021.2021<1/2020.2021
    ⇒1/2×2 + 1/3×3 + 1/4×4 +…..+1/2021×2021< 1/2.2+1/2.3+1/3.4+….+1/2020.2021 
                                                                          A<1/4+1/2-1/3+1/3-1/4+….+1/2020-1/2021
                                                                          A< 1/4+1/2-1/2021=3/4-1/2021<3/4 
    ⇒A <3/4(đccm)
      c) 1/2.2=1/2.2
        1/3.3=1/3.3
        1/4.4<1/3.4 
        ……….. 
     
      1/2021.2021<1/2020.2021
    ⇒1/2×2 + 1/3×3 + 1/4×4 +…..+1/2021×2021<1/2.2+1/3.3+1/3.4+…..+1/2020.2021
                                                                     A< 1/4+1/9+1/3-1/4+…….+1/2020.2021
                                                                     A< 13/36+1/3-1/2020.2021<13/36<13/18 
    ⇒A<13/18(đccm) 

  2. kymmailien
    kymmailien
    20 Tháng Một, 2023 at 9:36 sáng
    Reply
    a)Vì
    1/{2×2}< 1/{1×2}
    1/{3×3}< 1/{2×3}
    1/{4×4}< 1/{3×4}
    ….
    1/{2021×2021}< 1/{2020×2021}
    Nên:
    A<1/{1×2} + 1/{2×3}+ 1/{3×4}+ …+ 1/{2020×2021}
    A<2-1/{1×2} + 3-2/{2×3}+ 4-3/{3×4}+ …+ 2021-2020/{2020×2021}
    A<2/{1×2}- 1/{1×2} + 3/{2×3}- 2/{2×3}+ 4/{3×4}- 3/{3×4}+ …+ 2021/{2020×2021}- 2020/{20210×2021}
    A<1-1/2+1/2-1/3+1/3-1/4+…+1/2020-1/2021
    A<1-1/2021
    A<2020/2021
    Vì 2020/2021< 1
    Nên A<2020/2021 < 1
    B)Vì
    1/{2×2}= 1/{1×2}
    1/{3×3}< 1/{2×3}
    1/{4×4}< 1/{3×4}
    ….
    1/{2021×2021}< 1/{2020×2021}
    Nên:
    A<1/{2×2} + 1/{2×3}+ 1/{3×4}+ …+ 1/{2020×2021}
    A<1/4 + 3-2/{2×3}+ 4-3/{3×4}+ …+ 2021-2020/{2020×2021}
    A<1/4 + 3/{2×3}- 2/{2×3}+ 4/{3×4}- 3/{3×4}+ …+ 2021/{2020×2021}- 2020/{20210×2021}
    A<1/4 +1/2-1/3+1/3-1/4+…+1/2020-1/2021
    A<1/4+ 1/2 -1/2021
    A<3/4-2020/2021
    Vì 3/4- 1/2021< 3/4
    Nên A< 3/4
    c)Ta có:
    Vì
    1/{2×2}= 1/{2×2}
    1/{3×3}= 1/{3×3}
    1/{4×4}< 1/{3×4}
    ….
    1/{2021×2021}< 1/{2020×2021}
    Nên:
    A<1/{2×2} + 1/{3×3}+ (1/{3×4}+ …+ 1/{2020×2021})
    A<1/4+ 1/9+ (4-3/{3×4}+ …+ 2021-2020/{2020×2021})
    A<1/4+ 1/9+ (4/{3×4}- 3/{3×4}+ …+ 2021/{2020×2021}- 2020/{20210×2021})
    A<1/4+ 1/9+(1/3-1/4+…+1/2020-1/2021)
    A<1/4+ 1/9+1/3-1/2021
    A<13/36+1/3-1/2021
    A< 25/36-1/2021
    Ta có:
    25/36= 25/36
    13/18= {13×2}/{18×2}= 26/36
    Vì 25/36<26/36
    Nên 25/36< 13/18
    Nên 25/36- 1/2021< 13/18
    Mà A<25/36- 1/2021
    Nên A<13/18
    Vậy A< 13/18

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