If a, b, c, and d are in H.P., What can we say about the terms bcd, acd, abd, and abc

If a, b, c, and d are in Harmonic Progression (H.P.), it means that their reciprocals are in Arithmetic Progression (A.P.).

So, 1/a, 1/b, 1/c, and 1/d are in A.P.

Let's denote these terms as p, q, r, and s respectively. So, p = 1/a, q = 1/b, r = 1/c, and s = 1/d.

Now, we know that in an A.P., the product of means is equal to the product of extremes.

So, we have:

qr = ps 1/b * 1/c = 1/a * 1/d 1/(bc) = 1/(ad)

From this, we can say that the terms ad and bc are in A.P.

Similarly, we can prove that the terms bcd, acd, abd, and abc are also in A.P.

To do this, we can consider the terms bcd, acd, abd, and abc as the reciprocals of the terms 1/(bcd), 1/(acd), 1/(abd), and 1/(abc) respectively.

Since the terms a, b, c, and d are in H.P., their reciprocals are in A.P., and therefore the reciprocals of the terms bcd, acd, abd, and abc are also in A.P.

So, we can conclude that if a, b, c, and d are in H.P., then the terms bcd, acd, abd, and abc are in A.P.