Let 4X, 2X, X be the numbers.

Then, average of reciprocals of these numbers ( 1/4X + 1/2X + 1/X ) ÷ 3 = 7/72 (given in the problem).

That is, ( 1/4X + 2/4x + 4/4X ) ÷ 3 = 7/72.

That is, ( 7/4X) ÷ 3 = 7/72. Or, 7/4X =( 7×3)/72. That is, 7/4X =21/72 Or, 7/4X =7/24.

By equating the numerators and denominators of both sides, we have 7 = 7. And 4X = 24. Or, X = 6.

So, the first number = 4X = 4×6 = 24

Second number = 2X = 2×6 = 12

Third number = X = 6.

Checking for correctness :-

Numbers: 24, 12,& 6.

Average of sum of their reciprocals = {(1/24) + (1/12) +(1/6)} ÷ 3 ={(1/24) + (2/24) + (4/24)} ÷ 3

= (7/24) ÷ 3 = 7/(24×3) = 7/72 = value mentioned in the problem question given.