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prob25 - Problem 25 section 3.2 Ugh This problem Well lets...

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Problem 25, section 3 . 2 Ugh. This problem. Well, lets see. We want to find a sum for m summationdisplay k =0 floorleft k floorright For simplicity, we’ll let n = floorleft m floorright - 1, so that floorleft m floorright = n + 1. Why is n significant? Well, if you right out the sum m summationdisplay k =0 floorleft k floorright = 0 + 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + . . . you’ll notice that n is the second to last term that appears (before we start adding floorleft m floorright . This is important because we don’t know how many times we add floorleft m floorright . But we do now how many times we add everything below it. As you can see above, this fromula looks a lot like m k =0 (2 k + 1) k . But is isn’t quite the same, because we may not add floorleft m floorright as much as possible. Thus, we need to break this sum into two parts: n summationdisplay i =0 (2 k + 1) k + summationdisplay floorleft m floorright You’ll notice the sum on the left is relatively straight forward, and if you work it out, it is equal to n ( n +1)(2 n +1) 6 + n ( n +1) 2 . Thus we only need to figure
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