数列 $u_1$、$u_2$、$u_3$、$\ldots$ 满足 $u_1=4$，且当 $n\geqslant1$ 时，$u_{n+1}=3u_n-2$。

用数学归纳法证明：对所有正整数 $n$，$u_n=3^n+1$。

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。

验证首项，再假设第k项成立并代入递推式，推出第k+1项的目标形式。

For $n=1$, $3^1+1=4=u_1$, so the formula is true.

Assume that $u_k=3^k+1$ for some positive integer $k$. Using the recurrence,

Hence truth for $n=k$ implies truth for $n=k+1$. Since the formula holds for $n=1$, by mathematical induction

for every positive integer $n$.