If a,b,c are in Ap then show that a^2(b+c),b^2(c+a),c^2(a+b) are in Ap
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Answered by
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Step-by-step explanation:
As
a
,
b
,
c
are in A.P., we have
2
b
=
a
+
c
and hence
a
2
(
b
+
c
)
+
c
2
(
a
+
b
)
=
a
2
b
+
a
2
c
+
c
2
a
+
c
2
b
=
a
2
b
+
a
c
(
a
+
c
)
+
b
c
2
=
a
2
b
+
a
c
×
2
b
+
b
c
2
=
b
(
a
2
+
2
a
c
+
c
2
)
=
b
(
a
+
c
)
2
=
b
⋅
(
2
b
)
2
=
4
b
3
=
2
b
2
×
2
b
=
2
b
2
(
a
+
c
)
i.e.
a
2
(
b
+
c
)
+
c
2
(
a
+
b
)
=
2
b
2
(
a
+
c
)
or
c
2
(
a
+
b
)
−
b
2
(
a
+
c
)
=
b
2
(
a
+
c
)
−
a
2
(
b
+
c
)
i.e.
a
2
(
b
+
c
)
,
b
2
(
c
+
a
)
and
c
2
(
a
+
b
)
are in A.P.
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