The Five Positive Numbers

Algebra Level pending

Let a, b, c, d and e are real positive numbers, such that
{a, b, c, d, e} > 0 and \(a\neq b\neq c\neq d\neq e\).

Now consider following properties of {a, b, c, d, e}
1. a, b, c in A.P. (Arithmetic Progression)
2. b, c, d in G.P. (Geometric Progression)
3. c, d, e in A.P. (Arithmetic Progression)
4. \(a^{2}\) + e = [ a \(\times\) b ] + d
5. c + d + e = [ 2 \(\times\) \(a^{2}\) ]

If [ a + b + c + d + e ] is represented in terms of \(\frac{p}{q}\), where p and q are co-primes.
What is value of [ 3 \(\times\) q ] - p.

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