用数学归纳法证明：对所有正整数 $n$，$2025^n+47^n-2$ 能被 $46$ 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。

用归纳法，并把2025与47分别写成46的倍数加1。相邻两项之差明显含因子46。

Let $F(n)=2025^n+47^n-2$. For $n=1$,

so the result holds.

Assume that $F(k)$ is divisible by $46$ for some positive integer $k$. Then

This is a multiple of $46$. Since $F(k)$ is a multiple of $46$ by the hypothesis, $F(k+1)$ is also divisible by $46$.

Therefore, by mathematical induction, $\boxed{2025^n+47^n-2\text{ is divisible by }46}$ for every positive integer $n$.