Getting progressive with arithmetic, geometric and harmonicby K Sengupta Tags: arithmetic, geometric, harmonic, progressive 

#1
Sep3008, 10:10 AM

P: 110

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 (TP) 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 (TP) 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
Sep3008, 12:46 PM

P: 657

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 


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