Arithmetic Sequences, in Depth

For positive integers $n$ consider the sequence $$a_1 = 1, a_2 = \frac{1}{2}, a_3 = \frac{1+3}{2}, a_4 = \frac{1+3}{2+4}, a_5 = \frac{1+3+5}{2+4} \ldots$$ What is the minimum value of $n > 1$ such that

$$|f(n) - 1| \le \frac{1}{10}?$$

Source: Lexington Math Tournament