perplexus.info :: Sequences : Geometric, Arithmetic and Harmonic Harness

Geometric, Arithmetic and Harmonic Harness (Posted on 2014-10-19)

Each of A, B, C and D is a positive integer with A < B < C < D
having gcd(A, B, C, D) = 1 such that:

(i) A, B and C are in geometric sequence, and:
(ii) B, C and D are in arithmetic sequence, and:
(iii) A, B and D are in harmonic sequence.

Does there exist an infinite number of quadruplets satisfying the given conditions? Give reasons for your answer.

See The Solution Submitted by K Sengupta

Rating: 4.0000 (2 votes)

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Analytical solution (spoiler)

| Comment 1 of 3

My answer is NO, there are not an infinite number of quadruplets that satisfy. In fact, I only find one.

Here is my analysis:

a) C = B^2/A, because of the geometric sequence

b) D = 2B^2/A - B, because of the arithmetic sequence

c) Then, because of the harmonic sequence,

1/B - 1/A = (1-1/(2B^2/A - B)) - 1/B

Combining and eliminating fractions gives

(2A-B)(2B-A) = -A^2

then 3A^2 -5AB + 2B^2 = 0

then (A-B)(3A - 2B) = 0

d) A = B is not an allowed solution, so the only solution is 3A = 2B,

which gives rise to the lowest term integral solution of (4,6,9,12)

Posted by Steve Herman on 2014-10-19 09:36:51

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