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Points A1, A2,..., An have coordinates (xi, yi) for i = 1 to n. G1 divides A1A2 in ratio 1:1 internally; G2 divides G1A3 in ratio 1:2 internally; G3 divides G2A4 in ratio 1:3 internally; G4 divides G3A5 in ratio 1:4 internally; and so on until all n points are exhausted. Find the coordinates of the final point Gₙ₋₁.

1. (sum_(i=1)ⁿ i*xi / n, sum_(i=1)ⁿ i*yi / n)
2. (sum_(i=1)ⁿ xi / n, sum_(i=1)ⁿ yi / n)
3. (sum_(i=1)ⁿ (i-1)*xi / (n-1), sum_(i=1)ⁿ (i-1)*yi / (n-1))
4. (n * sum_(i=1)ⁿ xi, n * sum_(i=1)ⁿ yi)

Correct answer: (sum_(i=1)ⁿ xi / n, sum_(i=1)ⁿ yi / n)

Solution

By induction: Gₖ = average of A1 through Aₖ₊₁. So Gₙ₋₁ = (x1+x2+...+xn)/n = (sum xi)/n. Similarly for y-coordinates.

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