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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## Saturday, December 13, 2008

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