Let A_1,A_2,A_3,...,A_n are n Points in a plane whose coordinates are (x_1,y_1),(x_2,y_2),....,(x_n,y_n) respectively. A_1A_2 is bisected at the point P_1,P_1A_3 is divided in the ratio 1:2 at P_2,P_2A_4 is divided in the ratio 1:3 at P_3,P_3A_5 is divided in the ratio 1:4 at P_4 and the so on until all n points are exhausted. find the coordinates of the final point so obtained.

Let A_1,A_2,A_3,...,A_n are n Points in a plane whose coordinates are

1.	Let A_1,A_2,A_3,...,A_n are n Points in a plane whose coordinates are (x_1,y_1),(x_2,y_2),....,(x_n,y_n) respectively. A_1A_2 is bisected at the point P_1,P_1A_3 is divided in the ratio 1:2 at P_2,P_2A_4 is divided in the ratio 1:3 at P_3,P_3A_5 is divided in the ratio 1:4 at P_4 and the so on until all n points are exhausted. find the coordinates of the final point so obtained.
Answer» Solution :`P_1` is midpoint of `A_1A_2`. `THEREFORE""P_1-=((x_1+x_2)/(2),(y_1+y_2)/(2))` `P_2` DIVIDES `P_1A_3` in `1:2`. `therefore""P_2-=((2((x_1+x_2)/2)+x_3)/(2+1),(2((y_1+y_2)/2)+y_3)/(2+1))` `-=((x_1+x_2+x_3)/(3),(y_1+y_2+y_3)/(3))` Now, `P_3` divides `P_2A_4` in ` 1:3` `therefore""P_3-=((3.((x_1+x_2+x_3)/3)+x_4)/(3+1),(3.((y_1+y_2+y_3)/3)+y_4)/(3+1))` `-=((x_1+x_2+x_3+x_4)/(4),(y_1+y_2+y_3+y_4)/(4))` PROCEEDING in this manner, we get `P_n-=((x_1+x_2+x_3+....x_n)/(n),(y_1+y_2+y_3+....y_n)/(n))`.

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