2025 DSE 数学延伸部分-M2 真题 答案 详解

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答案(英文)

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The general term of $(1+5x)^n = C_r^n(5x)^r = C_r^n5^rx^r$

$a_1=27$

$1\times C_1^15^1+m\times1=27$

$m=27-5n$

$a_2=285$

$1\times C_2^15^2+m\times C_1^15^1=285$

$25\left[\frac{n(n-1)}{2}\right]+5(27-5n)n=285$

$5n^2-49n+114=0$

$(5n-19)(n-6)=0$

$n=\frac{19}{5}$ (rej.) or $6$

$m=27-5(6)$

$=-3$

Therefore,

$\begin{cases}m=-3\n=6\end{cases}$

P.1 (20)

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