[Coda]

Welcome to your first step into calculus! This stuff you're doing right now is the fundamentals that eventually lead into integrals (on the continuous side) and recurrence relations (on the discrete side, which you probably WON'T do in high school).

I can't quite tell what you're trying to do there. Are you trying to find the sum of a_n for n = 1 to ___? If so then your work looks right, although you could have made life a little easier for yourself by simplifying the expression before summing it:

1/(n+1) - 1/(n+2) = ((n+2) - (n+1))/((n+1)(n+2)) = 1/((n+1)(n+2))
n=1 -> 1/(2*3) = 1/6 = 10/60
n=2 -> 1/(3*4) = 1/12 = 5/60
n=3 -> 1/(4*5) = 1/20 = 3/60
n=4 -> 1/(5*6) = 1/30 = 2/60
Σa_n from 1 to 4 = 20/60 = 1/3

Same results!

I'm not sure what S_n there means, though, unless that's supposed to represent the closed form of the sum? I know how to do that but it's... a much more advanced technique than what I would have guessed to you to use at this level. (PROVING that a given formula -- if you already know it -- is the closed form of a sum is much easier.)