1. Product of A.P.? Sum of G.P.?

Algebra Level pending

In my time zone, (UTC +8), it's the night of the 5th day to 2015, so here comes the first question...

Given four terms \(a_1\), \(a_2\), \(a_3\) and \(a_4\).

Let \(a_1\), \(a_2\), \(a_3\) be a arithmetic progression with a product of 20. (i.e. \(a_1a_2a_3=20\))

Let \(a_2\), \(a_3\), \(a_4\) be a geometric progression with a sum of 15. (i.e. \(a_2+a_3+a_4=15\))

A closed form is pretty hard to get (even wolfram alpha spits at me...) so something else has to be asked...

Given that \(a_1\), \(a_2\), \(a_3\) and \(a_4\) are in the set of real numbers, and that \(a_1<a_2<a_3<a_4\), find the H.M. of \(a_1\), \(a_2\), \(a_3\) and \(a_4\).

Round the answer off to three significant decimal digits.

A closed form is appreciated (although most likely complicated).

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