proved that g(a2)=g(a-2)
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If one geometric mean G and two arithmetic means A
1
and A
2
be inserted between two given quantities, prove that G
2
=(2A
1
−A
2
)(2A
2
−A
1
).
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ANSWER
Let a and b be two given quantities. It is given that G is the geometric mean of a and b
Therefore, G=
ab
G
2
=ab
It is also given that A
1
,A
2
are two arithmetic means between a and b. Therefore, a,A
1
,A
2
,b is an A.P. with common difference d=
3
b−a
Therefore, A
1
=a+d=a+
3
b−a
=
3
2a+b
A
2
=a+2d=a+
3
2(b−a)
=
3
a+2b
So, 2A
1
−A
2
=2(
3
2a+b
)−(
3
a+2b
)=a
and 2A
2
−A
1
=2(
3
a+2b
)−(
3
2a+b
)=b
Therefore,
(2A
1
−A
2
)(2A
2
−A
1
)=ab
(2A
1
−A
2
)(2A
2
−A
1
)=G
2
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