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The formula for calculating the sum of all natural integers from 1 to n is well-known:
Similary, we know about the formula for calculating the sum of the first n squares:
Now, we reduce one of the two multipliers of each product by one to get the following sum:
Mn = 0 · 1 + 1 · 2 + 2 · 3 + 3 · 4 + ... + (n − 1) · n
Find an explicit formula for calculating the sum Mn.
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:
Here, we multiple the summands successively with 1, 2, 1, 2, 1, 2, ...
(a) Find an explicit formula to calculate this sum SN(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 Sn 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 Im and Ym for a given m.
Determine the growth behaviour by expressing Im and Ym with the big O notation
Find the smallest positive integer N that satisfies all of the following conditions:
• N is a square.
• N is a cube.
• N is an odd number.
• N is divisible by twelve prime numbers.
How many digits does this number N have?
Find the area of the surface that is generated by revolving the portion of the curve y=x^2 between x=0 and x=1 about the y-axis.
A a parabola having a vertex located at (-4,-8) interpreted the x-axis at x=-2 and x=-6. Determine the length of the arc of this parabola from the interpreted points.
Find the tangent to the parabola y2 = 6x − 3 perpendicular to the line x + 3y = 7
Find the tangent to the parabola y
2 = 6x − 3 perpendicular to the line x + 3y = 7
Find the rate at which the reciprocal of a number changes as the number increases.
Find the volume of the largest rectangular solid which can be inscribed in the
ellipsoid
x
2
a2
+
y
2
b
2
+
z
2
c
2
= 1
Uxx + Uyy =0 convert the situation equation into its Canonical form and find out its general solution
