Hexagon Line

Answer: 250


The sequence of hexagon lines
 
1-hexagon line has perimeter of 6
Each hexagon adds 4 to the perimeter
$\therefore$ perimeter = 6 + 4$\times$ number of hexagons added

So 1002 = 6 + 4$\times$?
$\Rightarrow$ 996 = 4$\times$?
996$\div$4 = 249

The perimeter is 1002 cm when 249 hexagons have been added to the first hexagon, so there are 250 hexagons altogether


Counting the lines in the pattern
End hexagons contribute 5 cm to perimeter
Non-end hexagons contribute 4 cm to perimeter

2 end hexagons contribute 10 cm to the perimeter
$\therefore$ the non-end hexagons contribute 1002$-$10 = 992 cm to the perimeter

992$\div$4 = 248
There are 248 non-end hexagons and 2 end hexagons = 250 hexagons in total


Alternatively, notice that each hexagon contributes $4$cm to the total perimeter, except the end two which contribute $2$ extra cm ($1$cm each), so if we take $2$cm off the total perimeter and divide by $4$ we will have the total number of hexagons in our shape. This gives us $n=250$.