Question

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 1

`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))`.