## Getting progressive with arithmetic, geometric and harmonic

Five positive integers P, Q, R, S and T, with P< Q < R <S < T, are such that:

(i) P, Q and R (in this order) are in arithmetic progression, and:
(ii) Q, R and S (in his order) are in geometric progression, and:
(iii) R, S and T (in this order) are in harmonic progression.

(I) Determine the minimum value of (T-P) such that there are precisely two quintuplets (P, Q, R, S, T) that satisfy all the given conditions.

(II) What is the minimum value of (T-P) such that there are precisely three quintuplets (P, Q, R, S, T) that satisfy all the given conditions?

Note: A harmonic progression is a progression whose reciprocals form an arithmetic progression. For example, {12, 15, 20,30} is harmonic since {1/12, 1/15, 1/20,1/30} is arithmetic.

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 I - Spoiler 162: (54,81,108,144,216) and (288,324,360,400,450) II - Spoiler 768: (96,192,288,432,864), (200,320,440,605,968), and (432,576,720,900,1200) DaveE