your perimeter is just the sum of all the sides. to figure out the arithmetic sequence here, you first need to list the perimeters of all the squares they gave you.
the first square has a perimeter of 4 -- each side is 1 unit long, and there are 4 sides. that leaves you with 4.
the second square has a perimeter of 8. each side is 2 units, and there are 4 sides. that leaves you with 8. repeat this logic and you get the sequence: 4, 8, 12, 16, 20.
observe those numbers and look for a common difference.
8 - 4 = 4
12 - 8 = 4
16 - 12 = 4
20 - 16 = 4
this tells you that, with each new figure, 4 is being added to the perimeter. the arithmetic sequence expressing this would read aₓ = x + 4.
for the second one, use the same process. count the number of squares and list them: 5, 9, 13, 17. this one isn't as easy to decode, but you'll note that there are 4 units being added each time, once again. sadly, they don't offer you something along the lines of the first problem. if n equals the picture number (aka the number in the sequence), you can see that the equation a(n) = 4n + 1 works.
because picture 1 is the first in the sequence: 4(1) + 1 = 5. picture 1 has 5 units.
because picture 2 is second: 4(2) + 1 = 9. picture 2 has 9 units.
because picture 3 is third: 4(3) + 1 = 13. picture 3 has 13 units. etc