### Question Description

## Explanation & Answer

1- On the first ring you have 1 hexagon,

on the second you have 6, on the third : 12, and on the 4th : 18

The total number of hexagons is 1+6+12+18=37

f(4)=3*4^2-3*4+1=3*16-12+1=37. The polynomial function f(r) gives the total number of hexagons when r=4

2- We have from the second ring an arithmetic suite u1=6 , u2=12, u3=18,

u4 should be 24 and so on. We are going to calculate the sum from u1 until u19 (20 rings including u0=1)

let's calculate u19=6+(19-1)*6=144

S=u1+...+u19=(u1+u19)*19/2=1140 and we have to add 1 hexagon for the 1st ring so the final result is 1141 hexagons.