If $0 < a < 1$, then the value of $a + \frac{1}{a}$ is

### Concept & Logic This problem relies on the foundational arithmetic inequality regarding a positive number and its reciprocal. For any positive real number $a$, the sum of the number and its reciprocal is always greater than or equal to 2. $$a + \frac{1}{a} \geq 2$$ Equality holds strictly when $a = 1$. ### Step-by-Step Solution Given the condition: $0 < a < 1$ This means $a$ is a positive fraction strictly less than 1. According to the Arithmetic Mean-Geometric Mean (AM-GM) inequality for positive numbers: $\frac{a + \frac{1}{a}}{2} \geq \sqrt{a \cdot \frac{1}{a}}$ $\frac{a + \frac{1}{a}}{2} \geq 1$ $a + \frac{1}{a} \geq 2$ Since the equality $a + \frac{1}{a} = 2$ only occurs when $a = 1$, and we are given that $a < 1$, the sum must be strictly greater than 2. ### Exam Strategy & Shortcut **Value Assumption:** Pick a simple fraction between 0 and 1. Let $a = 0.5$ (or $\frac{1}{2}$). Then, $\frac{1}{a} = 2$. Sum = $0.5 + 2 = 2.5$. 2.5 is clearly greater than 2. ### Common Pitfall Students might confuse the bounds and think that because $a$ is small (less than 1), the overall sum will also be small. They forget that as $a$ gets smaller, its reciprocal $\frac{1}{a}$ gets exponentially larger, driving the sum well above 2. ### Final Answer Therefore, the correct answer is **greater than 2**.