Bunuel
OK. Harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.
Arithmetic progression: 5, 9, 13, 17, ...
Harmonic progression: 1/5, 1/9, 1/13/ 1/17, ...
General form of harmonic progression is: a, a/(1+d), a/(1+2d), a/(1+3d)...
We have the sequence: 3, 4, 6. So, we should check if they can fit in the above form:
Assume that 3 is the first term (in my previous post I assumed that 3 was just nth term, but it's the same) --> if 3 is the first term then a=3 --> next term 4 should be: 4=3/(1+d), from this equation d=-1/4 --> if a=3 and d=-1/4 and sequence 3, 4, 6 do form the harmonic progression then the third term must be calculated by the formula a/(1+2d) and this result must be 6. Let's check a/(1+2d)=3/(1-2*1/4)=6.
Hence we can conclude that 3, 4, 6 form the harmonic progression.
Hope now it's clear.
Can you please provide the source of this question?
Well simple way of solving this can be to check if the reversal of the given numbers are in AP.
For 3,4,6 to be in HP : 1/3 , 1/4 , 1/6 should be in AP.
for a,b,c to be in AP b = [a+b]/2
so for 3,4,6 to be in HP
1/4 = {1/3 + 1/6}/2 which satisfies. SO the three numbers are in HP!!!