To solve the problem, we need to analyze the quadratic equation given by:
\[ a_1x^2 - a_2x + a_3 = 0 \]
where \( a_1, a_2, a_3 \) are natural numbers (i.e., \( a_1, a_2, a_3 \in \mathbb{N} \)). We are tasked with finding the least value of \( a_1 \) such that the equation has two distinct real roots in the interval \( (1, 2) \).
### Step 1: Conditions for Distinct Real Roots
For the quadratic equation to have distinct real roots, the discriminant must be positive. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac = (-a_2)^2 - 4a_1a_3 = a_2^2 - 4a_1a_3 \]
Thus, we require:
\[ a_2^2 - 4a_1a_3 > 0 \tag{1} \]
### Step 2: Roots in the Interval (1, 2)
Let the roots be \( \alpha \) and \( \beta \). By Vieta's formulas, we know:
- \( \alpha + \beta = \frac{a_2}{a_1} \)
- \( \alpha \beta = \frac{a_3}{a_1} \)
Since both roots lie in the interval \( (1, 2) \), we can derive the following inequalities:
1. \( 1 < \alpha < 2 \)
2. \( 1 < \beta < 2 \)
From the sum of the roots:
\[ 2 < \alpha + \beta < 4 \]
This gives us:
\[ 2 < \frac{a_2}{a_1} < 4 \]
Multiplying through by \( a_1 \) (which is positive), we have:
\[ 2a_1 < a_2 < 4a_1 \tag{2} \]
From the product of the roots:
\[ 1 < \alpha \beta < 4 \]
This gives us:
\[ 1 < \frac{a_3}{a_1} < 4 \]
Multiplying through by \( a_1 \):
\[ a_1 < a_3 < 4a_1 \tag{3} \]
### Step 3: Finding the Least Value of \( a_1 \)
Now we have inequalities (1), (2), and (3) to satisfy. We will analyze these inequalities to find the least value of \( a_1 \).
From inequality (2):
- \( a_2 \) must be at least \( 2a_1 + 1 \) (since \( a_2 \) is a natural number).
From inequality (3):
- \( a_3 \) must be at least \( a_1 + 1 \) (since \( a_3 \) is also a natural number).
Let’s assume \( a_1 = 5 \):
- Then \( a_2 \) must satisfy \( 10 < a_2 < 20 \) (so \( a_2 \) can be 11, 12, ..., 19).
- \( a_3 \) must satisfy \( 5 < a_3 < 20 \) (so \( a_3 \) can be 6, 7, ..., 19).
Now, we check the discriminant condition (1):
For \( a_1 = 5 \):
- Choose \( a_2 = 11 \) and \( a_3 = 6 \):
\[
D = 11^2 - 4 \cdot 5 \cdot 6 = 121 - 120 = 1 > 0
\]
Thus, the discriminant is positive, confirming that the roots are distinct.
### Conclusion
The least value of \( a_1 \) that satisfies all conditions is:
\[
\boxed{5}
\]
