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: S (2) n = 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 S (2) n . (Assume that n is a multiple of 2.) Now, we consider the sum S (3) n : S (3) n = 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,
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:
S
(2)
n = 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 S
(2)
n . (Assume that n is a multiple of 2.)
Now, we consider the sum S
(3)
n :
S
(3)
n = 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 S
(3)
n . (Assume that n is a multiple of 3.)
(c) Express S
(3)
n in the form of
S
(3)
n = 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 S
(m)
n . (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)
n = Im · Sn/m − Ym · 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 S, of the positive integers from 1 to n was
already part of Problem A.3. For this problem, we consider the following rollercoaster sum:
se) = 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 S). (Assume that n is a multiple of 2.)
Now, we consider the sum S):
S) = 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 S. (Assume that n is a multiple of 3.)
(c) Express S in the form of
S9) = I . Sn/3 - Y •n
where S, is the formula from Problem A.3 and I, Y are rational constants.
(d) Find a formula for the general case of S). (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
Sm) = Im · Sn/m – Ym · 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5fbe1b3-0e67-4dfb-bd73-75e01a369440%2Fcbf579af-9d4f-4bfb-b942-00c1e6ebc015%2Fxiz01z8_processed.png&w=3840&q=75)
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