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Saturday, December 13, 2008

Q) Find all three-term Harmonic Progressions a, b, c of strictly increasing positive integers in which a = 20, and b divides c .
( from RMO - 2008)


A) Let c = k.b ......... where k is a natural number

From the definition of a harmonic progression:
1/a + 1/c = 2/b

(1/20) + (1/bk) = 2/b
40 - (20/k) = b
Using values of k E [1, 20] we get b = 20, 30, 35, 36, 38, 39. But exclude 20 as we require strictly increasing values (since a = 20 already, b must not be 20).

[ Remember -
use only those values of k which are factors of 20, else you won't get integral values... so k E {1, 2, 4, 5, 10, 20} ]

Then using :
c = 20 / [(40/b) - 1]
we get c = 60, 140, 180, 380, 780 respectively for the aforementioned values of b (excluding 20).


So there are 5 such Harmonic Progressions :

20, 30,60
20, 35, 140
20, 36, 180
20, 38, 380
20, 39, 780

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