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`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
`A_(1)A_(2)` is bisected at `G_(1),G_(1)A_(3)` is divided in the ratio `1:2 at G_(2),G_(2) A_(4)` is divided in the ratio `1:3` at `G_(3)`. The process is continued until all n points are exhausted then find the co ordinates of the final point `G_(n)`.

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The correct Answer is:
To find the coordinates of the final point \( G_n \) after the process described in the question, we can follow these steps: ### Step 1: Find the coordinates of \( G_1 \) The point \( G_1 \) is the midpoint of the segment \( A_1A_2 \). The coordinates of \( G_1 \) can be calculated using the midpoint formula: \[ G_1 = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] ### Step 2: Find the coordinates of \( G_2 \) The point \( G_2 \) divides the segment \( G_1A_3 \) in the ratio \( 1:2 \). Using the section formula, the coordinates of \( G_2 \) are given by: \[ G_2 = \left( \frac{1 \cdot x_3 + 2 \cdot \frac{x_1 + x_2}{2}}{1 + 2}, \frac{1 \cdot y_3 + 2 \cdot \frac{y_1 + y_2}{2}}{1 + 2} \right) \] This simplifies to: \[ G_2 = \left( \frac{x_3 + x_1 + x_2}{3}, \frac{y_3 + y_1 + y_2}{3} \right) \] ### Step 3: Find the coordinates of \( G_3 \) The point \( G_3 \) divides the segment \( G_2A_4 \) in the ratio \( 1:3 \). Again, using the section formula: \[ G_3 = \left( \frac{1 \cdot x_4 + 3 \cdot \frac{x_3 + x_1 + x_2}{3}}{1 + 3}, \frac{1 \cdot y_4 + 3 \cdot \frac{y_3 + y_1 + y_2}{3}}{1 + 3} \right) \] This simplifies to: \[ G_3 = \left( \frac{x_4 + x_3 + x_1 + x_2}{4}, \frac{y_4 + y_3 + y_1 + y_2}{4} \right) \] ### Step 4: Generalize for \( G_n \) Continuing this process, we can observe a pattern. The coordinates of \( G_n \) can be expressed as: \[ G_n = \left( \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n}, \frac{y_1 + y_2 + y_3 + \ldots + y_n}{n} \right) \] ### Step 5: Final coordinates of \( G_n \) Thus, the final coordinates of \( G_n \) can be expressed as: \[ G_n = \left( \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n}, \frac{y_1 + y_2 + y_3 + \ldots + y_n}{n} \right) \]
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