Toán Lớp 8: Cho 1/a + 1/b + 1/c = 1/(a + b + c) CM: 1/a^2021 + 1/b^2021 + 1/c^2021 = 1/(a^2021 + b^2021 + c^2021) (a,b,c khác 0, a + b + c khác 0)

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Toán Lớp 8:  Cho 1/a + 1/b + 1/c = 1/(a + b + c) CM: 1/a^2021 + 1/b^2021 + 1/c^2021 = 1/(a^2021 + b^2021 + c^2021) (a,b,c khác 0, a + b + c khác 0)

trucoanh55 · Answer

Giải đáp:     Ta c&oacute;: 1/a + 1/b + 1/c = 1/(a + b + c)   -> 1/a + 1/b = 1/(a + b + c) - 1/c   -> (a + b)/ab = {c - (a + b + c)}/{c.(a + b + c)}   -> (a + b).c.(a + b + c) = ab.(-a - b) = -ab.(a + b)   -> (a + b).c.(a + b + c) + ab.(a + b) = 0   &harr; (a +b).(ca + cb +c^2 + a^2) = 0   &harr; (a + b).[a.(b + c) + c.(b + c)] = 0   &harr; (a + b).(b + c).(c + a) = 0   &harr; $ left[ begin{matrix} a + b = 0   b + c = 0    c + a = 0 end{matrix} right.$   &harr; $ left[ begin{matrix} a = -b   b = -c    c = -a end{matrix} right.$   &middot;) Nếu a = -b   &rarr; a^2021 = -b^2021   &rarr; 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   -> 1/a^2021 + 1/b^2021 + 1/c^2021 = 1/(a^2021 + b^2021 + c^2021)   &middot;) Nếu b = -c   &rarr; b^2021 = -c^2021   &rarr; 1/b^2021 + 1/c^2021 + 1/a^2021 = 1/(-c^2021) + 1/c^2021 + 1/a^2021 = 1/a^2021   -> 1/(b^2021 + c^2021 +a^2021) = 1/(-c^2021 +c^2021 + a^2021) = 1/a^2021   -> 1/a^2021 + 1/b^2021 + 1/c^2021 = 1/(a^2021 + b^2021 + c^2021)   &middot;) Nếu c = -a   &rarr; c^2021 = -a^2021   &rarr; 1/c^2021 + 1/a^2021 + 1/b^2021 = 1/(-a^2021) + 1/a^2021 + 1/b^2021 = 1/b^2021   -> 1/(c^2021 + a^2021 +b^2021) = 1/(-a^2021 +a^2021 + b^2021) = 1/b^2021   -> 1/a^2021 + 1/b^2021 + 1/c^2021 = 1/(a^2021 + b^2021 + c^2021)    ∘ dariana