# Getting progressive with arithmetic, geometric and harmonic

1. Sep 30, 2008

### K Sengupta

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.

2. Sep 30, 2008

### davee123

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

DaveE