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Byju's Answer
Standard XII
Mathematics
Sum of n Terms
If there are ...
Question
If there are
(
2
n
+
1
)
terms in an AP, then show that :
S
u
m
o
f
o
d
d
t
e
r
m
s
S
u
m
o
f
e
v
e
n
t
e
r
m
s
=
n
+
1
n
.
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Solution
Let
a
be the first term and
d
be the common difference of the given AP.
Now, sum of odd terms =
S
1
=
a
1
+
a
3
+
a
5
+
.
.
.
.
.
+
a
2
n
+
1
S
1
=
n
+
1
2
(
a
1
+
a
2
n
+
1
)
S
1
=
n
+
1
2
[
a
1
+
a
1
+
(
2
n
+
1
−
1
)
d
]
S
1
=
n
+
1
2
(
a
+
a
+
2
n
d
)
=
(
n
+
1
)
(
a
+
n
d
)
Also, sum of even terms =
S
2
=
a
2
+
a
4
+
a
6
+
.
.
.
.
+
a
2
n
S
2
=
n
2
(
a
2
+
a
2
n
)
S
2
=
n
2
[
(
a
+
d
)
+
a
+
(
2
n
−
1
)
d
]
S
2
=
n
2
(
2
a
+
2
n
d
)
=
n
(
a
+
n
d
)
Therefore,
S
1
S
2
=
(
n
+
1
)
(
a
+
n
d
)
n
(
a
+
n
d
)
=
n
+
1
n
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0
Similar questions
Q.
If there are (2n+1) terms in AP, then prove that the ratio of the sum of odd terms and the sum of even terms is (n+1) : n.
Q.
If
m
th term of an AP is
1
n
and
n
th is
1
m
then show that
(
m
n
)
t
h
term of an AP is
1
.
Q.
If there are
2
n
+
1
terms in A.P., then prove that the ratio of the sum of odd terms and even terms is
(
n
+
1
)
:
n
Q.
if the sum of first n terms of an AP is 2n^2-n+1, then find the tenth term of the AP
Q.
If
a
n
+
b
n
a
n
−
1
+
b
n
−
1
is the A.M. between a, b then find the value of n.
OR
If there are
(
2
n
+
1
)
terms in A.P., then prove that the ratio of the sum of the odd terms and the even terms is
(
n
+
1
)
:
n
.
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