Further Geometric Numbers (2) (Posted on 2007-03-19)

	Submitted by K Sengupta
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Let the terms of the geometric sequence be b, br and b*r^2; where b is positive.Otherwise, since r is greater than 1, the three terms of the sequence cannot correspond to the sides of a triangle whenever b<=0 If, r is not an integer, then r> 1, and so, the length of the greatest side of the triangle must correspond to b*r^2. Now, the sum of lengths of two sides of the triangle must exceed that of the third side, and so: b+br> b*r^2 Or, r^2 - r -1 <0 Or, (1- sqrt(5))/2 <r < (1+ sqrt(5))/2.....(#) Or, 1< r<(1+ sqrt(5))/2 Or, [r] = 1 Also, from (#): -(1+ sqrt(5))/2 <-r < -1 Or, [-r] = -2 Consequently: [r] + [-r] = 0 = 1-2 = -1 However, if r is an integer, then by (#): r< 2, so that r = 1, giving: [r] + [-r] = 0 Consequently, [r] + [-r] is always 1, whenever r is not an integer. But, [r] + [-r] is not equal to 1 whenever r is an integer.

	Subject	Author	Date
	Solution	Praneeth	2007-12-27 03:17:25
	Solution	Bractals	2007-03-19 13:07:28
	re: Quick spoiler	Old Original Oskar!	2007-03-19 08:47:32
	Quick spoiler	e.g.	2007-03-19 08:43:47