`A_(1)(x_(1),y_(1)),A_(2)(x_(2),y_(2)),A_(3)(x_(3),y_(3))`……….are a point in a plane such that
If `x_(1),y_(1)=2` and `x_(i)` 's form a G.P. with common ratio 2 and `y_(i)` 's form a G.P. with common ratio 3, then find the co ordinates of G.

Answer

To find the coordinates of the centroid \( G \) of the points \( A_1(x_1, y_1), A_2(x_2, y_2), A_3(x_3, y_3), \ldots \) given the conditions in the problem, we will follow these steps:

### Step 1: Determine the coordinates of the points based on the given conditions.

We know that:
- \( A_1(x_1, y_1) = (2, 2) \)
- The \( x_i \) coordinates form a geometric progression (G.P.) with a common ratio of 2.
- The \( y_i \) coordinates form a geometric progression (G.P.) with a common ratio of 3.

Using the formula for the \( n \)-th term of a G.P., we can express the coordinates as follows:

For \( x_i \):
- \( x_1 = 2 \)
- \( x_2 = 2 \cdot 2 = 4 \)
- \( x_3 = 2 \cdot 2^2 = 8 \)
- In general, \( x_n = 2 \cdot 2^{n-1} = 2^n \)

For \( y_i \):
- \( y_1 = 2 \)
- \( y_2 = 2 \cdot 3 = 6 \)
- \( y_3 = 2 \cdot 3^2 = 18 \)
- In general, \( y_n = 2 \cdot 3^{n-1} \)

### Step 2: Find the coordinates of the centroid \( G \).

The coordinates of the centroid \( G \) of \( n \) points \( (x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n) \) are given by:

\[ G\left( \frac{x_1 + x_2 + \ldots + x_n}{n}, \frac{y_1 + y_2 + \ldots + y_n}{n} \right) \]

#### Calculate \( x_1 + x_2 + \ldots + x_n \):

\[ x_1 + x_2 + \ldots + x_n = 2 + 4 + 8 + \ldots + 2^n \]

This is a geometric series with:
- First term \( a = 2 \)
- Common ratio \( r = 2 \)
- Number of terms \( n \)

The sum of the first \( n \) terms of a G.P. is given by:

\[ S_n = a \frac{r^n - 1}{r - 1} = 2 \frac{2^n - 1}{2 - 1} = 2(2^n - 1) = 2^{n+1} - 2 \]

Thus,

\[ x_1 + x_2 + \ldots + x_n = 2^{n+1} - 2 \]

#### Calculate \( y_1 + y_2 + \ldots + y_n \):

\[ y_1 + y_2 + \ldots + y_n = 2 + 6 + 18 + \ldots + 2 \cdot 3^{n-1} \]

This is also a geometric series with:
- First term \( a = 2 \)
- Common ratio \( r = 3 \)
- Number of terms \( n \)

The sum of the first \( n \) terms is:

\[ S_n = a \frac{r^n - 1}{r - 1} = 2 \frac{3^n - 1}{3 - 1} = 2 \frac{3^n - 1}{2} = 3^n - 1 \]

Thus,

\[ y_1 + y_2 + \ldots + y_n = 3^n - 1 \]

### Step 3: Substitute into the centroid formula.

Now substituting back into the centroid formula:

\[ G\left( \frac{2^{n+1} - 2}{n}, \frac{3^n - 1}{n} \right) \]

### Final Answer:

The coordinates of the centroid \( G \) are:

\[ G\left( \frac{2^{n+1} - 2}{n}, \frac{3^n - 1}{n} \right) \]