perplexus.info :: Just Math : M-D Sequences

M-D Sequences (Posted on 2014-11-09)

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

No Solution Yet Submitted by Bractals

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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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