The well-known formula for calculating the sum Sn of the positive integers from 1 to n...

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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) 1·1+2·2+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 S²). (Assume that n is a multiple of 2.) Now, we consider the sum S(3): S(3) = 1·1+2·2+3·3+1 4+2 5+3·6+...+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 S³). (Assume that n is a multiple of 3.) (c) Express S³) 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. = . 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) 1·1+2·2+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 S²). (Assume that n is a multiple of 2.) Now, we consider the sum S(3): S(3) = 1·1+2·2+3·3+1 4+2 5+3·6+...+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 S³). (Assume that n is a multiple of 3.) (c) Express S³) 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. = .

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a To find an explicit formula for the sum Sn2 we observe that the pattern of the summands is repeating after every two terms 11 22 13 24 15 26 and so on We can split the sum into two separate sums one ... View the full answer