Question :

If one AM ‘a’ and two GM’s p and q are inserted between any two given numbers then find value of 2A
Submitted on 10/7/2024 | Answered by Vandana Rana

Answer :

Let the numbers be
x
and
y


A
is their
A
.
M
.


2
A
=
x
+
y.......


(1)


Also
p
,
q
are two geometric means between
x
and
y


∴

x
,
p
,
q
,
y
are in
G
.
P
.


Since
x
,
p
,
q
are in
G
.
P
.


∴
p2

=
x
q

or x=
p2/q.......(2)




Since
p
,
q
,
y
are in
G
.
P
.


∴
q2

=
p
y


y= q2/p.....
....(3)
Adding (2)and (3)

x+y =p2/q +q2/p

x+y =( p3 +q3)/pq
put it in (1)


2A =(p3 +q3)/pq