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NDA Math Sequences and Series
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© examsnet.com
Question : 26
Total: 79
Let
S
n
be the sum of the first n terms of an AP. If
S
2
n
=
3
n
+
14
n
2
, then what is the common difference?
[NDA II 2019]
5
6
7
9
Validate
Solution:
Given that,
S
2
n
=
3
n
+
14
n
2
When
n
=
n
2
then
S
n
=
3
n
2
+
14
n
2
4
⇒
S
n
=
3
n
2
+
7
n
2
2
⇒
S
n
=
1
2
n
(
3
+
7
n
)
∴
S
n
−
1
=
1
2
(
n
−
1
)
[
3
+
7
(
n
−
1
)
]
⇒
S
n
−
1
=
1
2
(
n
−
1
)
[
3
+
7
n
−
7
]
=
1
2
(
n
−
1
)
(
7
n
−
4
)
∵
T
n
=
S
n
−
S
n
−
1
=
1
2
n
(
3
+
7
n
)
−
1
2
(
n
−
1
)
(
7
n
−
4
)
⇒
T
n
=
1
2
[
3
n
+
7
n
2
−
7
n
2
+
4
n
+
7
n
−
4
]
⇒
T
n
=
1
2
(
14
n
−
4
)
=
7
n
−
2
⇒
T
1
=
7
×
1
−
2
=
5
⇒
T
2
=
7
×
2
−
2
=
12
∴ the common difference
T
2
−
T
1
=
12
−
5
=
7
i.e.,
d
=
7
© examsnet.com
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