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NIMCET 2008 Question Paper with solutions
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© examsnet.com
Question : 13
Total: 120
If
H
1
,
H
2
,
.
.
.
,
H
n
are
n
harmonic means between a and b ,
a
≠
b
,
then the value of
H
1
+
a
H
1
−
a
+
H
n
+
b
H
n
−
b
is equal to
n
+
1
n
−
1
2
n
2
n
+
3
Validate
Solution:
a.
H
1
,
H
2
,
.
.
.
,
H
n
,
b
in HP.
⇒
1
a
,
1
H
1
,
1
H
2
.
.
.
1
H
n
1
b
in AP.
⇒
1
b
=
1
a
+
(
n
+
2
−
1
)
d
⇒
1
b
−
1
a
=
(
n
+
1
)
d
⇒
d
=
1
b
−
1
a
(
n
+
1
)
∴
1
H
1
=
1
a
+
d
=
1
a
+
1
b
−
1
a
n
+
1
=
1
a
+
a
−
b
a
b
(
n
+
1
)
=
b
(
n
+
1
)
+
a
−
b
a
b
(
n
+
1
)
⇒
H
1
=
a
b
(
n
+
1
)
b
n
+
a
Using componendo and Dividendo, we get
⇒
H
n
+
b
H
n
−
b
=
a
n
+
a
+
a
n
+
b
a
n
+
a
−
a
n
−
b
=
a
+
b
+
2
b
n
b
−
a
.....(i)
Also,
1
H
n
=
1
b
−
d
=
1
b
−
a
−
b
a
b
(
n
+
1
)
=
a
(
n
+
1
)
−
a
+
b
a
b
(
n
+
1
)
⇒
H
n
=
a
b
(
n
+
1
)
a
n
+
b
⇒
H
n
b
=
a
n
+
a
a
n
+
b
Using componendo and Dividendo, we get
⇒
H
n
+
b
H
n
−
b
=
a
n
+
a
+
a
n
+
b
a
n
+
a
−
a
n
−
b
=
a
+
b
+
2
a
n
a
−
b
.....(ii)
On adding Eqs. (i) and (ii),
∴
H
1
+
a
H
1
−
a
+
H
n
+
b
H
n
−
b
=
a
+
b
+
2
b
n
b
−
a
−
a
+
b
+
2
a
n
b
−
a
=
2
n
(
b
−
a
)
b
−
a
=
2
n
© examsnet.com
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