# 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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