`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)=a,y_(1)=b` and `x_(i)`s form an A.P. with common difference 2 and `y_(i)` 's form an A.P with common difference 4,then find the co-ordinates of G, the centroid.

Answer

To find the coordinates of the centroid \( G \) of the points \( A_1(x_1, y_1), A_2(x_2, y_2), A_3(x_3, y_3), \ldots, A_n(x_n, y_n) \), we follow these steps:

### Step 1: Define the coordinates of the points
Given:
- \( x_1 = a \)
- \( y_1 = b \)

The \( x_i \) coordinates form an arithmetic progression (A.P.) with a common difference of 2:
- \( x_2 = x_1 + 2 = a + 2 \)
- \( x_3 = x_2 + 2 = a + 4 \)
- Continuing this pattern, we have:
\[ x_n = a + 2(n - 1) \]

The \( y_i \) coordinates form an A.P. with a common difference of 4:
- \( y_2 = y_1 + 4 = b + 4 \)
- \( y_3 = y_2 + 4 = b + 8 \)
- Continuing this pattern, we have:
\[ y_n = b + 4(n - 1) \]

### Step 2: Write the coordinates of all points
The coordinates of the points can be summarized as:
- \( A_1(a, b) \)
- \( A_2(a + 2, b + 4) \)
- \( A_3(a + 4, b + 8) \)
- ...
- \( A_n(a + 2(n - 1), b + 4(n - 1)) \)

### Step 3: Calculate the centroid coordinates
The coordinates of the centroid \( G \) are given by:
\[ G_x = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]
\[ G_y = \frac{y_1 + y_2 + y_3 + \ldots + y_n}{n} \]

#### Calculate \( G_x \):
\[ G_x = \frac{a + (a + 2) + (a + 4) + \ldots + (a + 2(n - 1))}{n} \]
This can be simplified:
\[ G_x = \frac{na + (0 + 2 + 4 + \ldots + 2(n - 1))}{n} \]
The sum \( 0 + 2 + 4 + \ldots + 2(n - 1) \) is an arithmetic series with \( n \) terms, first term 0, and last term \( 2(n - 1) \):
\[ \text{Sum} = \frac{n}{2} \times (0 + 2(n - 1)) = n(n - 1) \]
Thus,
\[ G_x = \frac{na + n(n - 1)}{n} = a + (n - 1) \]

#### Calculate \( G_y \):
\[ G_y = \frac{b + (b + 4) + (b + 8) + \ldots + (b + 4(n - 1))}{n} \]
This can be simplified:
\[ G_y = \frac{nb + (0 + 4 + 8 + \ldots + 4(n - 1))}{n} \]
The sum \( 0 + 4 + 8 + \ldots + 4(n - 1) \) is also an arithmetic series:
\[ \text{Sum} = \frac{n}{2} \times (0 + 4(n - 1)) = 2n(n - 1) \]
Thus,
\[ G_y = \frac{nb + 2n(n - 1)}{n} = b + 2(n - 1) \]

### Final Coordinates of the Centroid
Therefore, the coordinates of the centroid \( G \) are:
\[ G = (a + (n - 1), b + 2(n - 1)) \]