用数学归纳法证明：对所有正整数 $n$，$6^{4n}+38^n-2$ 都能被 $74$ 整除。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

先验证n=1。归纳时比较相邻两个表达式，差可提取37；括号内为偶数，才确保差能被74整除。

Let $F_n=6^{4n}+38^n-2$. For $n=1$,

so the result is true.

Assume that $F_k$ is divisible by $74$ for some positive integer $k$. Then

Since $k\geqslant1$, both $6^{4k}$ and $38^k$ are even. The bracket is therefore even, so the difference is divisible by $74$.

By the hypothesis $F_k$ is divisible by $74$, and thus so is $F_{k+1}$. The result holds for $n=1$ and truth for $n=k$ implies truth for $n=k+1$. Hence, by induction, $6^{4n}+38^n-2$ is divisible by $74$ for every positive integer $n$.