請參考附件
第 10 題
答案更改為 10^(2025/2)
114 成德高中
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114 成德高中
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Re: 114 成德高中
第 10 題
設 A = 1 + 2 + ... + 2024
B = a_1 * a_2 * ... * a_2025 = a_1^2025 * r^A
C = 1 + r + r^2 + ... + r^2024
D = a_1 + a_2 + ... + a_2025 = a_1C
a_1^a_1 * a_2^a_2 * ... * a_2025^a_2025 = 10^2 = 100
a_1^a_1 * (a_1r)^(a_1r) * ... * (a_1r^2024)^(a_1r^2024) = 100 ... (1)
a_1^a_2025 * a_2^a_2024 * ... * a_2025^a_1 = 10^(-1) = 1/10
a_1^(a_1r^2024) * (a_1r)^(a_1r^2023) * ... * (a_1r^2024)^a_1 = 1/10 ... (2)
(1) * (2)
a_1^(2a_1C) * r^(2024a_1C) = 10
(a_1^2 * r^2024)^(a_1C) = 10
所求 = B^D = (a_1^2025 * r^A)^(a_1C)
= (a_1^2 * r^2024)^[(2025/2)(a_1C)]
= 10^(2025/2)
設 A = 1 + 2 + ... + 2024
B = a_1 * a_2 * ... * a_2025 = a_1^2025 * r^A
C = 1 + r + r^2 + ... + r^2024
D = a_1 + a_2 + ... + a_2025 = a_1C
a_1^a_1 * a_2^a_2 * ... * a_2025^a_2025 = 10^2 = 100
a_1^a_1 * (a_1r)^(a_1r) * ... * (a_1r^2024)^(a_1r^2024) = 100 ... (1)
a_1^a_2025 * a_2^a_2024 * ... * a_2025^a_1 = 10^(-1) = 1/10
a_1^(a_1r^2024) * (a_1r)^(a_1r^2023) * ... * (a_1r^2024)^a_1 = 1/10 ... (2)
(1) * (2)
a_1^(2a_1C) * r^(2024a_1C) = 10
(a_1^2 * r^2024)^(a_1C) = 10
所求 = B^D = (a_1^2025 * r^A)^(a_1C)
= (a_1^2 * r^2024)^[(2025/2)(a_1C)]
= 10^(2025/2)