用数学归纳法证明：对于所有正整数 $n$，$7^{2n}+97^n-50$ 都能被 $48$ 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。

相邻指数项分别带出49与97，与原表达式相减后得到48与96两个倍数。

Let $F_n=7^{2n}+97^n-50$. For $n=1$,

so the statement holds.

Assume $F_k$ is divisible by $48$ for some positive integer $k$. Then

which is divisible by $48$. The hypothesis therefore implies that $F_{k+1}$ is divisible by $48$.

Since the statement holds for $n=1$ and truth for $n=k$ implies truth for $n=k+1$, the result follows by induction for every positive integer $n$.