The acceleration due to gravity becomes `(g)/(16)` at a height of nR from the surface of earth. Find the value of n.

Explanation

To solve the problem, we need to find the value of \( n \) such that the acceleration due to gravity at a height of \( nR \) from the surface of the Earth is \( \frac{g}{16} \).

### Step-by-Step Solution:

1. **Understanding the Formula for Acceleration Due to Gravity**:
The acceleration due to gravity \( g \) at a distance \( r \) from the center of the Earth is given by:
\[ g = \frac{GM}{r^2} \]
where \( G \) is the universal gravitational constant and \( M \) is the mass of the Earth.

2. **Acceleration Due to Gravity at Height \( h \)**:
At a height \( h \) above the surface of the Earth, the formula for acceleration due to gravity becomes:
\[ g' = \frac{GM}{(R + h)^2} \]
where \( R \) is the radius of the Earth.

3. **Substituting the Height**:
We are given that the height is \( h = nR \). Substituting this into the formula gives:
\[ g' = \frac{GM}{(R + nR)^2} = \frac{GM}{((1 + n)R)^2} \]

4. **Setting Up the Equation**:
We know that at this height, the acceleration due to gravity is \( \frac{g}{16} \). Therefore, we can set up the equation:
\[ \frac{GM}{((1 + n)R)^2} = \frac{g}{16} \]

5. **Substituting \( g \)**:
Since \( g = \frac{GM}{R^2} \), we can substitute \( g \) in the equation:
\[ \frac{GM}{((1 + n)R)^2} = \frac{GM/R^2}{16} \]

6. **Canceling \( GM \)**:
We can cancel \( GM \) from both sides (assuming \( GM \neq 0 \)):
\[ \frac{1}{((1 + n)R)^2} = \frac{1}{16R^2} \]

7. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 16R^2 = ((1 + n)R)^2 \]

8. **Simplifying the Equation**:
Dividing both sides by \( R^2 \) (assuming \( R \neq 0 \)):
\[ 16 = (1 + n)^2 \]

9. **Taking the Square Root**:
Taking the square root of both sides gives:
\[ 4 = 1 + n \quad \text{or} \quad -4 = 1 + n \quad (\text{we discard this as } n \text{ must be positive}) \]

10. **Solving for \( n \)**:
From \( 4 = 1 + n \), we find:
\[ n = 4 - 1 = 3 \]

### Final Answer:
Thus, the value of \( n \) is \( \boxed{3} \).