All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
Further Geometric Numbers (2) (Posted on 2007-03-19) Difficulty: 1 of 5
If three successive terms of a geometric sequence with ratio r correspond to the lengths of the three sides of a triangle, then determine whether or not [r]+[-r]=-1.

[x] is the greatest integer ≤ x.

  Submitted by K Sengupta    
No Rating
Solution: (Hide)
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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolutionPraneeth2007-12-27 03:17:25
SolutionSolutionBractals2007-03-19 13:07:28
Solutionre: Quick spoilerOld Original Oskar!2007-03-19 08:47:32
SolutionQuick spoilere.g.2007-03-19 08:43:47
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (12)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information