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

Toán Lớp 7: Chứng minh rằng : ` 1/3 + 2/3^2 + 3/3^3 +4/3^4 + … + 100/3^100 < 3/4 `

Tháng Hai 20, 2023 Toán học 2 Comments 0 views

Toán Lớp 7: Chứng minh rằng :
1/3 + 2/3^2 + 3/3^3 +4/3^4 + … + 100/3^100 < 3/4

Comments ( 2 )

  1. lanthuhoa
    lanthuhoa
    20 Tháng Hai, 2023 at 9:21 chiều
    Reply

    *-Gửi bạn-*
    toan-lop-7-chung-minh-rang-1-3-2-3-2-3-3-3-4-3-4-100-3-100-3-4

  2. ngoclanquynh
    ngoclanquynh
    20 Tháng Hai, 2023 at 9:20 chiều
    Reply
                        Giải :
    Đặt A = 1/3 + 2/(3^2) + 3/(3^3) + 4/(3^4) + … + 100/(3^100)
    ⇒ 3A = 1 + 2/3 + 3/(3^2) + 4/(3^3) + …. + 100/(3^99)
    ⇒ 2A = 1 + (2/3 – 1/3) + [3/(3^2) – 2/(3^2)] +…+ [100/(3^99) – 99/(3^99)] – 100/(3^100)
    ⇒ 2A = 1+ 1/3 + 1/(3^2) + 1/(3^3) +…+ 1/(3^99) – 100/(3^100)
    Đặt B = 1/3 + 1/(3^2) + 1/(3^3) +…+ 1/(3^99)
    ⇒ 3B = 1 + 1/3 + 1/(3^2) + 1/(3^3) +…+ 1/(3^98)
    ⇒ 2B = 1 – 1/(3^99)
    ⇒ B = [1 – 1/(3^99)]/2
    Thay vào 2A
    ⇒ 2A = 1 + [1 – 1/(3^99)]/2 – 100/(3^100) = 1+ 1/2 – 1/(2×3^99) – 100/(3^100) < 1+ 1/2 = 3/2
    ⇒ A < 3/4 

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

Khánh Ngân

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