ISL 1970 Problem 19

Let $1<n\in\mathbb{N}$ and $1\le a\in\mathbb{R}$ and there are $n$ number of $x_i, i\in\mathbb{N}, 1\le i\le n$ such that $x_1=1$ and $\frac{x_{i}}{x_{i-1}}=a+\alpha _ i$ for $2\le i\le n$ , where $\alpha _i\le \frac{1}{i(i+1)}$ . Prove that $\sqrt[n-1]{x_n}< a+\frac{1}{n-1}$ .