The well-known formula for calculating the sum Sn of the positive integers from 1 to n was already part of Problem A.3. For this problem, we consider the following rollercoaster sum: S2) 11+22+1 3+2 4+...+ 1 (n 1) + 2 n Here, we multiple the summands successively with 1, 2, 1, 2, 1, 2, ... (a) Find an explicit formula to calculate this sum S). (Assume that n is a multiple of 2.) Now, we consider the sum S(3): S(3) = 11+22+33+1 4+2 5+36+...+1 (n 2) + 2 (n 1) +3.n - Here, we multiple the summands successively with 1, 2, 3, 1, 2, 3, ... (b) Again, find an explicite formula for the sum S). (Assume that n is a multiple of 3.) (c) Express S) in the form of S(3) = = I Sn/3 - Y n where Sn is the formula from Problem A.3 and I, Y are rational constants. (d) Find a formula for the general case of Sm). (That means we multiple the summands successively with 1, 2, 3, ..., m, 1, 2, 3, ..., m, ...; Assume that n is a multiple of m.) (e) Now, express the general formula as S(m) Im Sn/m - Ym n n and find explicit equations to calculate Im and Ym for a given m. (f) Determine the growth behaviour by expressing Im and Ym with the big O notation. = .