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Answers:Interesting! First, let's think about the graph of the function y = sqrt(x). And let's imagine a second graph, with straight lines between each of the integer square roots. i.e. the graph contains the points (0,0), (1,1), (4,2), (9,3), (16,4), (25,5) etc, with straight lines connecting these points. Now if the square root function really did follow the straight line function, then all the sums you compute would have remainder 0. For instance, between 4 and 5, the sum of the remainders would be 1/9 + 2/9 + 3/9 + 4/9 + 5/9 + 6/9 + 7/9 + 8/9 = 36/9 = 4.0 exactly. What this suggests is that the extra remainder you get (1/6) is the difference between the square root function and the straight line function. It is possible to calculate the area underneath the two curves and confirm this is the case. Between the points (a^2, a) and ((a+1)^2, a+1) (for some arbitary number a), the area under the square root curve can be calculated using calculus: area = (integate from a^2 to (a+1)^2) x^1/2 dx = [from a^2 to (a+1)^2] 2/3 x^3/2 = 2/3 ((a+1)^3 - a^3) = 2a^2 + 2a + 2/3 And the area under the straight line section is (using * for multiply): = width * (height of rectangle + 1/2 height of triangle) ((a+1)^2 - a) * (a + 1/2) = (2a + 1) * (a + 1/2) = 2a^2 + 2a + 1/2 So the difference between the two curves is 1/6. That's really quite an interesting result: even when a is very large (and the area between the curves is very long in the x direction and very small in the y direction) the difference in area is always the same. When you add up the remainders of each square root, you are in effect approximating the area of this difference curve. As the number a gets larger, this approximation gets more and more accurate.
Answers:Find a pattern: Even integers- -8, -6, -4, -2, 0, 2, etc... Difference between integers is two. So if "i" was any integer, the next integer would be "i+2" right? and the next next one would be "i+4" right?
Answers:The "principal" square root of a number is the positive square root. The negative is also a square root. So the answers are: The other square root is -2.1 (you had that). The given positive number is 2.1 ^ 2 (where ^ = squared) or 2.1 ^ 2 = 4.41 so the Given Positive Number is 4.41 2. I'm going to use ^.5 to mean square root here. What you have is 3(30^.5) I think. Or, in english, 3 times the square root of 30. The square root of 30 is 5.477, so 3 * 5.477 = 16.43 So, 16.43 lies between what 2 consecutive integers? The answer is 16 and 17 since 16.43 is greater than 16 and less than 17.
Answers:2n is an even integer. 2n + 2 and 2n + 4 are the next two consecutive even integers. 2n + (2n + 2) + (2n + 4) = 54 6n + 6 = 54 6n = 48 n = 8 The integers are 16, 18, and 20.