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M-D Sequences (Posted on 2014-11-09) Difficulty: 3 of 5


Let the sequence of real numbers { rk } be defined by
   rk = ak                                     if k = 1

      = ak*[ 1 - ( ak/[ 2*rk-1 ] )2 ]           if k > 1.
Prove that { rk } is a
strictly monotonically decreasing sequence with
   ak > rk > 0                                 for k > 1,
if the sequence of real numbers { ak } is a
monotonically decreasing sequence with
   ak > 0                                      for k ≥ 1.

No Solution Yet Submitted by Bractals    
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Question What am I missing? | Comment 1 of 3
Let a(k) = 1.  This qualifies as a monotonically decreasing sequence.

Then r(k) = 1 - 1/(2*r(k-1))^2

r(1) = 1
r(2) = 1 - 1/(2)^2 = 3/4
r(3) = 1 - 1/(3/2)^2 = 5/9
r(4) = 1 - 1/(10/9)^2 = 19/100
r(5) = 1 - 1/(19/50)^2, which is negative

So it appears that the theorem is false.

Did I do something wrong?

  Posted by Steve Herman on 2014-11-09 11:28:14
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