Answer to Question #268053 in Calculus for nikita

Question

The well-known formula for calculating the sum S_n of the positive integers from 1 to n was already part of Problem A.3. For this problem, we consider the following rollercoaster sum:

Here, we multiple the summands successively with 1, 2, 1, 2, 1, 2, ...

(a)  Find an explicit formula to calculate this sum S_N⁽²⁾. (Assume that n is a multiple of 2.) Now, we consider the sum:

Here, we multiple the summands successively with 1, 2, 3, 1, 2, 3, ...

(b) Again, find an explicite formula for the sum   . (Assume that n is a multiple of 3.)

(c)  Express in the form of

where S_n is the formula from Problem A.3 and I, Y are rational constants.

(d) Find a formula for the general case of . (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

and find explicit equations to calculate I_m and Y_m for a given m.

Determine the growth behaviour by expressing I_m and Y_m with the big O notation