Given that `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 `A :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 so on until all `n` points are exhausted. Find the final point so obtained.

Explanation

`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))`.