Three terms in Harmonic Progression. If the third terms decreas by four it is Arithmetic Progression. If the third terms decrease by 3 they are in Geometric Progression. Find the three terms.

Let the HP be 1/a ; 1/b; 1/c
So a,b,c are in AP (i)

If third term of the HP is decreased by 4, i.e. 1/c-4, it makes an AP (ii)
If third term of the HP is decreased by 3, i.e. 1/c-3, it makes an AP (iiI)

for the I, II, III eqns, form 3eqns in 3 variables and solve them to get the answer.

Let the HP be 1/a ; 1/b; 1/c
So a,b,c are in AP (i)

If third term of the HP is decreased by 4, i.e. 1/c-4, it makes an AP (ii)
If third term of the HP is decreased by 3, i.e. 1/c-3, it makes an AP (iiI)

for the I, II, III eqns, form 3eqns in 3 variables and solve them to get the answer.