A circular disk of unit radius is filled with a number of smaller circular disks arranged in the form of hexagon. Let `A_n` denotes a stack of disks arranged in the shape of a hexagon having 'n' disks on a side. The figure shows the configuration `A_3`. If A be the area of large disk, `S_n` be the number of disks in `A_n` configuration and `r_n` be the radius of each disk in `A_n` configuration, then `lim_(n->oo)(S_n)/n^2` `lim_(n->oo) nr_n`

Explanation

n,(n+1),(n+2),(n+3)...(2n-1)...n

`S_n=sum_(r=0)^(n-2)2(n+r)+n+n-1`

`=2sum_(r=0)^(n-2)+2sum_(r=0)^(n-2)r+2n-1`

`=2n^2-2n+n^2-3n+2+2n-1`

`S_n=3n^2-3n+1`

`lim_(n->oo)S_n/n^2`

`lim_(n->oo)3-3/n+1/n^2`

`3`.