Answer:
See below
Explanation:
we want to prove that A.M, G.M. and H.M between any two unequal positive numbers satisfy the following relations.
- (G.M)²= (A.M)×(H.M)
- A.M>G.M>H.M
well, to do so let the two unequal positive numbers be
where:
the AM,GM and HM of
and
is given by the following table:
Proof of I:
simplify addition:
reduce fraction:
simplify complex fraction:
reduce fraction:
rewrite:
hence, PROVEN
Proof of II:
square root both sides:
isolate right hand side expression to left hand side and change its sign:
square both sides:
expand using (a-b)²=a²-2ab+b²:
move -2√x_1√x_2 to right hand side and change its sign:
divide both sides by 2:
again,
expand:
move the middle expression to right hand side and change its sign:
cross multiplication:
hence,
PROVEN