Answer:
3rd and 5th option; s = [tex]n^{2}[/tex] + n and s = (n)(n + 1)
Step-by-step explanation:
There are many different methods, but since this is a multiple choice, the most direct solution is to just go down and try all of them.
The first option is s = 2n. Plugging in 1, you get 2, which works. However, plugging in 2 only gives 4, but you need six. s = 2n is not right.
s = [tex]n^{2}[/tex]: plugging in 1 already doesn't work, because [tex]1^{2}[/tex] = 1 ≠ 2.
s = [tex]n^{2}[/tex] + n: [tex]1^{2}[/tex] + 1 = 2 check, [tex]2^{2}[/tex] + 2 = 6 check, [tex]3^{2}[/tex] + 3 = 12 check! Bingo! That's an answer.
s= [tex]n^{2}[/tex] + 1: [tex]1^{2}[/tex] + 1 = 2, so far so good, but [tex]2^{2}[/tex] + 1 = 5 ≠ 6, so this is not right either.
Finally, s = (n)(n +1) definitely works because if you distribute it, you get [tex]n^{2}[/tex] + n, which we've already proven works. However, plugging numbers back in also works. For example, (2)(2 + 1) = 2 * 3 = 6, check. (3)(3 + 1) = 3 * 4 = 12, check!
Therefore, the two answers are s = [tex]n^{2}[/tex] + n and s = (n)(n + 1).
