Proposition to prove:
if a, b, c are in hp show that 1/a + 1/(b+c), 1/b + 1/(c+a), 1/c + 1/(a+b) are
also in hp.
Two things we must know
1. Three terms p,q,r are in hp if and only if their reciprocals
1/p, 1/q, 1/r are in ap.
2. Three terms u,v,w are in ap if and only if v-u = w-v
if a, b, c are in hp
That is true if and only if
1/a, 1/b, 1/c are in ap.
That is true if and only if
(1)
1/a + 1/(b+c), 1/b + 1/(c+a), 1/c + 1/(a+b) are also in hp
That will be true true if and only if after getting LCD's, we can show that:
, , and
are in hp. That will be true if and only if
, , and
are in ap.
They have the same denominators so they will be in ap if and only if their
numerators are in ap. That will be true if and only if
That will be true if and only if
That will be true if and only if
That will be true if and only if we can divide through by abc
and this will be true:
That will be true if and only if we can cancel and get
That will be true if and only if we multiply through by -1, and get
That will be true if and only if we can reverse the terms and get
And we have that that is true in equation (1) above. So the proposition is
proved.
Edwin