The problem asks for the value of 'n' in the molecular formula $\mathrm{Be_n Al_2Si_6O_{18}}$. This compound is a neutral silicate mineral known as beryl. To find the value of 'n', we will use the principle of charge neutrality, which states that the sum of the oxidation states of all atoms in a neutral compound must be zero.

1. Assign Oxidation States: First, we assign the common oxidation states to the known elements in the formula:

   • Beryllium (Be) is an alkaline earth metal (Group 2) and has an oxidation state of $+2$.
   • Aluminum (Al) is in Group 13 and has an oxidation state of $+3$.
   • Silicon (Si) in silicates has an oxidation state of $+4$.
   • Oxygen (O) in oxides and silicates has an oxidation state of $-2$.

2. Set up the Charge Balance Equation: The sum of the total positive charges from the cations (Be, Al, Si) must equal the sum of the total negative charges from the anions (O). Let the oxidation state of an element be denoted by O.S. $(n \times \text{O.S. of Be}) + (2 \times \text{O.S. of Al}) + (6 \times \text{O.S. of Si}) + (18 \times \text{O.S. of O}) = 0$

3. Substitute the Oxidation States into the Equation: Substitute the known oxidation states into the equation: $(n \times (+2)) + (2 \times (+3)) + (6 \times (+4)) + (18 \times (-2)) = 0$

4. Solve for n: Now, we solve the linear equation for 'n'. $2n + 6 + 24 - 36 = 0$ $2n + 30 - 36 = 0$ $2n - 6 = 0$ $2n = 6$ $n = \frac{6}{2}$ $n = 3$

5. Conclusion: The value of 'n' is 3. The molecular formula for beryl is $\mathrm{Be_3 Al_2Si_6O_{18}}$. We can verify the charge balance: Total positive charge = $3(+2) + 2(+3) + 6(+4) = 6 + 6 + 24 = +36$. Total negative charge = $18(-2) = -36$. The net charge is $+36 - 36 = 0$.