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標題:
數學 : 求值
發問:
若 a ≠ b , b ≠ c , c ≠ a 且 a2 = bc 及 b2 = ca 1) a + b + c 的值為何? 2) (1/a + 1/b + 1/c)(a2o11 + b? + c?) 的值為何? 更新: a , b , c 未必是實數。
最佳解答:
1) a2 - b2 = bc - ca (a - b)(a + b) = c(b - a) a + b = -c a + b + c = 0 b) (1/a + 1/b + 1/c)a2011 = a2010 + a2011 (b + c)/(bc) = a2010 + a2011 (-a)/a2 = a2010 - a2010 = 0 (1/a + 1/b + 1/c)b8 = b7 + b8 (a + c)/(ac) = b7 - b8 (-b)/b2 = b7 - b7 = 0 (1/a + 1/b + 1/c)c7 = c6 + c7 (a + b)/(ab) = c6 - c7 (-c)/c2 = c6 - c6 = 0 所以, (1/a + 1/b + 1/c)(a2011 + b8 + c7) = 0 2011-08-07 14:48:01 補充： c^2 = ab 的證明: a^2 = bc b^2 = ac a^2 b^2 = abc^2 ab = c^2
數學 : 求值
發問:
若 a ≠ b , b ≠ c , c ≠ a 且 a2 = bc 及 b2 = ca 1) a + b + c 的值為何? 2) (1/a + 1/b + 1/c)(a2o11 + b? + c?) 的值為何? 更新: a , b , c 未必是實數。
最佳解答:
1) a2 - b2 = bc - ca (a - b)(a + b) = c(b - a) a + b = -c a + b + c = 0 b) (1/a + 1/b + 1/c)a2011 = a2010 + a2011 (b + c)/(bc) = a2010 + a2011 (-a)/a2 = a2010 - a2010 = 0 (1/a + 1/b + 1/c)b8 = b7 + b8 (a + c)/(ac) = b7 - b8 (-b)/b2 = b7 - b7 = 0 (1/a + 1/b + 1/c)c7 = c6 + c7 (a + b)/(ab) = c6 - c7 (-c)/c2 = c6 - c6 = 0 所以, (1/a + 1/b + 1/c)(a2011 + b8 + c7) = 0 2011-08-07 14:48:01 補充： c^2 = ab 的證明: a^2 = bc b^2 = ac a^2 b^2 = abc^2 ab = c^2
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