Geometric to Arithmetic Progression

Algebra Level 3

In a sequence of four positive numbers, the first three are in geometric progression and the last three are in arithmetic progression. The first number is \(12\) and the last number is \( \frac{45}{2} \). The sum of the two middle numbers can be written as \(\frac{a}{b} \) where \(a\) and \(b\) are coprime positive integers. Find \(a+b\).

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