Refer to the
earlier problem.
Can you find all possible
non zero real quadruplet(s) (P, Q, R, S) that satisfy the following system of
simultaneous equations?
2*Q = P + 19/P, and:
2*R = Q + 19/Q, and:
2*S = R + 19/R, and:
2*P = S + 19/S.
Bonus Question:
In the problem given above, if the number
19 was replaced throughout by a
real constant M > 1, what would have been the possible non zero real quadruplet(s)
(P, Q, R, S) that satisfy the given set of equations,
in terms of M?
I just recognized the process for iterating to find the square root of M.
Define a Series S(n)
Let S(n+1) = (S(n) + M/S(n))/2.
As long as S(1) <> 0, this converges very quickly to sqrt(M).
If s(1) <> sqrt(M), then |S(n+1) - sqrt(M)| < |S(n) - sqrt(M)| for n > 2.
(If s(1) < 1 and M > 1, then this inequality does not hold for n = 1, but it holds thereafter. For instance, if M = 16 and s(1) = .25, then the series is
1 0.25
2 32.125
3 16.31152724
4 8.646214322
5 5.248367593
6 4.148467273
7 4.002656708
8 4.000000882
and each term starting with the 3rd is between 4 and the term before it)
So what does this have to with our problem?
Well, if P*P > M, then
P*P > Q*Q > R*R > S*S > P*P, which is a contradiction
Similarly, if P*P < M, then
P*P < Q*Q < R*R < S*S < P*P, which is a contradiction
So, the only solution is the one I have already given,
P = Q = R = S = +/- sqrt(M)
And yes, I know I haven't done the algebra, but I could and it would work.
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Or, to be more clear, if P*P <> M,
then Q*Q is between P*P and M,
and R*R is between Q*Q and M,
and S*S is between R*R and M,
and P*P is between S*S and M,
but this is a contradiction, so P*P = M
Edited on May 28, 2008, 5:11 pm
Edited on May 29, 2008, 9:31 am
Aha! (spoiler)
