When `n` number of particles each of mass `m` are at distances `x_(1)=a, x_(2)=ar, x_(3)=ar^(2)…….x_(n)=ar^(n)` units from origin on the x-axis, then the distance of their centre of mass from origin.

Explanation

`x_(cm)=(ma+m(ar)+m(ar^(2))+…..+m(ar^(n)))/(m+m+m+…..+m(n"terms"))`
`x_(cm)=(m(a+ar+ar^(2)+…+ar^(n)))/(mn)`
If `r gt 1` then `x_(cm)=(1)/(n)[(a(r^(n)-1))/(r-1)]=(a(r^(n-1)))/(n(r-1))`
If `r lt 1` then `x_(cm)=(1)/(n)[(a(1-r^(n)))/(1-r)]=(a(1-r^(n)))/(n(1-r))`