Toán cho a^2+b^2+c^2=1,a^3+b^3+c^3=1.Tính a^5+b^5+c^5 06/09/2021 By Melody cho a^2+b^2+c^2=1,a^3+b^3+c^3=1.Tính a^5+b^5+c^5
\[\begin{array}{l}{a^2} + {b^2} + {c^2} = {a^3} + {b^3} + {c^3} = 1\\ \to {a^2}(1 – a) + {b^2}(1 – b) + {c^2}(1 – c) = 0\\{a^2} + {b^2} + {c^2} = 1 \to {a^2} \le 1 \to \left| a \right| \le 1 \to (1 – a) \ge 0,1 – b \ge 0;1 – c \ge 0\\ \to {a^2}(1 – a) + {b^2}(1 – b) + {c^2}(1 – c) \ge 0 \to (a,b,c) = (1,0,0);(0,1,0);(0,0,1)\\ \to {a^5} + {b^5} + {c^5} = 1 + 0 + 0 = 1\end{array}\] Trả lời
\[\begin{array}{l}
{a^2} + {b^2} + {c^2} = {a^3} + {b^3} + {c^3} = 1\\
\to {a^2}(1 – a) + {b^2}(1 – b) + {c^2}(1 – c) = 0\\
{a^2} + {b^2} + {c^2} = 1 \to {a^2} \le 1 \to \left| a \right| \le 1 \to (1 – a) \ge 0,1 – b \ge 0;1 – c \ge 0\\
\to {a^2}(1 – a) + {b^2}(1 – b) + {c^2}(1 – c) \ge 0 \to (a,b,c) = (1,0,0);(0,1,0);(0,0,1)\\
\to {a^5} + {b^5} + {c^5} = 1 + 0 + 0 = 1
\end{array}\]
Đáp án:
a^5+b^5+c^5=1
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