Infinite number of bodies, each of mass `2kg`, are situated on `x`-axis at distance `1m,2m,4m,8m.........` respectively, from the origin. The resulting gravitational potential the to this system at the origing will be

Options

A. `-G`
B. `-(8//3)G`
C. `-(4//3)G`
D. `-4G`

Explanation

To solve the problem of finding the gravitational potential at the origin due to an infinite number of bodies each of mass \(2 \, \text{kg}\) situated on the x-axis at distances \(1 \, \text{m}, 2 \, \text{m}, 4 \, \text{m}, 8 \, \text{m}, \ldots\), we can follow these steps:

### Step 1: Understand the formula for gravitational potential
The gravitational potential \(V\) at a point due to a mass \(m\) at a distance \(r\) is given by:
\[ V = -\frac{Gm}{r} \]
where \(G\) is the gravitational constant.

### Step 2: Identify the distances and masses
The masses are \(m = 2 \, \text{kg}\) and the distances from the origin are \(r_n = 1, 2, 4, 8, \ldots\). These distances can be expressed as \(r_n = 2^{n-1}\) for \(n = 1, 2, 3, \ldots\).

### Step 3: Calculate the total gravitational potential at the origin
The total gravitational potential at the origin due to all the masses can be expressed as:
\[ V_{\text{total}} = \sum_{n=1}^{\infty} -\frac{G \cdot 2}{r_n} \]
Substituting \(r_n\):
\[ V_{\text{total}} = \sum_{n=1}^{\infty} -\frac{G \cdot 2}{2^{n-1}} \]

### Step 4: Factor out constants
We can factor out the constants from the summation:
\[ V_{\text{total}} = -2G \sum_{n=1}^{\infty} \frac{1}{2^{n-1}} \]

### Step 5: Recognize the summation as a geometric series
The series \(\sum_{n=1}^{\infty} \frac{1}{2^{n-1}}\) is a geometric series with the first term \(a = 1\) and common ratio \(r = \frac{1}{2}\). The sum of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{1}{1 - \frac{1}{2}} = 2 \]

### Step 6: Substitute back into the potential equation
Now substituting back into the potential equation:
\[ V_{\text{total}} = -2G \cdot 2 = -4G \]

### Conclusion
Thus, the resulting gravitational potential at the origin due to this infinite system of bodies is:
\[ \boxed{-4G} \]