A.P-G.P

Given that a1,a2,a3a_1,a_2,a_3 is an arithmetic progression in that order so that a1+a2+a3=15a_1+a_2+a_3=15 and b1,b2,b3b_1,b_2,b_3 is a geometric progression in that order so that b1b2b3=27b_1b_2b_3=27.

If a1+b1,a2+b2,a3+b3a_1+b_1, a_2+b_2, a_3+b_3 are positive integers and form a geometric progression in that order, determine the maximum possible value of a3a_3.

The answer is of the form a+bcd\dfrac{a+b\sqrt{c}}{d}, where aa, bb, cc, and dd are positive integers and the fraction is in its simplest form and cc is square free. Submit the value of a+b+c+d a + b + c + d .

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