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

Toán Lớp 8: Cho 1/a+1/b+1/c=1/(a+b+c) Chứng minh 1/a^2021 +1/b^2021 + 1/c^2021=1/(a^2021+b^2021+c^2021)

Tháng Hai 22, 2023 Toán học 0 Comments 0 views

Toán Lớp 8: Cho 1/a+1/b+1/c=1/(a+b+c)
Chứng minh 1/a^2021 +1/b^2021 + 1/c^2021=1/(a^2021+b^2021+c^2021)

Comments ( 1 )

  1. hongocha12
    hongocha12
    22 Tháng Hai, 2023 at 2:02 sáng
    Reply
    $\frac{1}{a}$+$\frac{1}{b}$+$\frac{1}{c}$=$\frac{1}{a+b+c}$
    ⇔$\frac{bc}{abc}$+$\frac{ca}{abc}$+$\frac{ab}{abc}$=$\frac{1}{a+b+c}$
    ⇔$\frac{ab+bc+ca}{abc}$=$\frac{1}{a+b+c}$
    ⇔(ab+bc+ca)(a+b+c)=abc
    ⇔ab(a+b)+bc(a+b)+ca(a+b)+abc+bc²+c²a-abc=0
    ⇔ab(a+b)+bc(a+b)+ca(a+b)+c²(a+b)=0
    ⇔(a+b)(ab+bc+ca+c²)=0
    ⇔(a+b)[b(a+c)+c(a+c)]=0
    ⇔(a+b)(a+c)(b+c)=0
    ⇔a+b=0⇔a=-b
    b+c=0⇔b=-c
    a+c=0⇔a=-c
    Trường hợp 1: a=-b 
    1/a^2021 +1/b^2021 + 1/c^2021
    =1/(-b)^2021 + 1/b^2021 + 1/c^2021
    =1/c^2021
    1/(a^2021+b^2021+c^2021)
    =1/(-b)^2021+b^2021+c^2021
    =1/c^2021
    Còn lại chứng minh tương tự…

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

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