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Byju's Answer
Standard XII
Mathematics
Geometric Progression
If the pth,...
Question
If the
p
t
h
,
q
t
h
,
r
t
h
terms of a H.P. be
a
,
b
,
c
respectively, prove that
(
q
−
r
)
b
c
+
(
r
−
p
)
c
a
+
(
p
−
q
)
a
b
=
0
.
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Solution
Given that
a
,
b
,
c
are the
p
t
h
,
q
t
h
.
r
t
h
terms of a H.P
1
a
,
1
b
,
1
c
are the
p
t
h
,
q
t
h
.
r
t
h
terms of an A.P
Let the A.P have
x
as the first term and
d
as the common difference
⇒
1
a
=
x
+
(
p
−
1
)
d
----1
⇒
1
b
=
x
+
(
q
−
1
)
d
----2
⇒
1
c
=
x
+
(
r
−
1
)
d
----3
1-2 we get
1
a
−
1
b
=
(
p
−
q
)
d
d
=
b
−
a
a
b
(
p
−
q
)
Similarly
2-3 we get
d
=
c
−
b
b
c
(
q
−
r
)
3-1 we get
d
=
a
−
c
a
c
(
r
−
p
)
⇒
d
=
b
−
a
a
b
(
p
−
q
)
=
c
−
b
b
c
(
q
−
r
)
=
a
−
c
a
c
(
r
−
p
)
⇒
We know that if
x
=
a
b
=
c
d
=
e
f
then
x
=
a
+
c
+
e
b
+
d
+
f
⇒
d
=
b
−
a
+
c
−
b
+
a
−
c
a
b
(
p
−
q
)
+
b
c
(
q
−
r
)
+
a
c
(
r
−
p
)
⇒
d
=
0
a
b
(
p
−
q
)
+
b
c
(
q
−
r
)
+
a
c
(
r
−
p
)
⇒
(
a
b
(
p
−
q
)
+
b
c
(
q
−
r
)
+
a
c
(
r
−
p
)
)
d
=
0
⇒
a
b
(
p
−
q
)
+
b
c
(
q
−
r
)
+
a
c
(
r
−
p
)
=
0
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0
Similar questions
Q.
If
p
t
h
,
q
t
h
,
r
t
h
terms of an HP. be
a
,
b
,
c
respectively, prove that
(
q
−
r
)
b
c
+
(
r
−
p
)
a
c
+
(
p
−
q
)
a
b
=
0
Q.
If the pth, qth and rth terms of an A.P. be a, b and c respectively, then prove that
a
(
q
−
r
)
+
b
(
r
−
p
)
+
c
(
p
−
q
)
=
0
.
Q.
If the
p
t
h
,
q
t
h
,
r
t
h
terms of a G.P. be a, b,c respectively prove that
a
q
−
r
b
r
−
p
c
p
−
q
=
1
Q.
If
p
t
h
term,
q
t
h
term and
r
t
h
term of H.P. are
a
,
b
and
c
respectively.
Then prove that
b
c
(
q
−
r
)
+
c
a
(
r
−
p
)
+
a
b
(
p
−
q
)
=
0
Q.
If
a
,
b
,
c
are
p
t
h
,
q
t
h
and
r
t
h
terms respectively of an A.P prove that:
p
(
b
−
c
)
+
q
(
c
−
a
)
+
r
(
a
−
b
)
=
0
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