please give correct answer
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Answer:
The given series is,1
2
+(1
2
+2
2
)+(1
2
+2
2
+3
2
)+...
a
n
=(1
2
+2
2
+3
2
+...+n
2
)
=
6
n(n+1)(2n+1)
=
6
n(2n
2
+3n+1)
=
6
2
3
+3n
2
+n
=
3
1
n
3
+
2
1
n
2
+
6
1
n
∴S
n
=
k=1
∑
n
a
k
=
k=1
∑
n
(
3
1
k
2
+
2
1
k
2
+
6
1
k)
=
3
1
k=1
∑
n
k
3
+
2
1
k=1
∑
n
k
2
+
36
1
k=1
∑
n
k
=
3
1
(2)
2
n
2
(n+1)
2
+
2
1
×
6
n(n+1)(2n+1)
+
6
1
×
2
n(n+1)
=
6
n(n+1)
[
2
n(n+1)
+
2
(2n+1)
+
2
1
]
=
6
n(n+1)
[
2
n
2
+n+2n+1+1
]
=
6
n(n+1)
[
2
n
2
+n+2n+2
]
=
6
n(n+1)
[
2
n(n+1)+2(n+1)
]
=
6
n(n+1)
[
2
(n+1)(n+2)
]
=
12
n(n+1)
2
(n+2)
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