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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 103
Total: 106
Find the sum of the following series up to n terms :
13
1
+
13+23
1+3
+
13+23+33
1+3+5
 + ...
Solution:  
We have
13
1
+
13+23
1+3
+
13+23+33
1+3+5
 + ...
∴ n th term is
an =
13+23+...+n3
1+3+5+...to‌n‌terms
an =
[
n(n+1)
2
]2
n
2
[2+(n−1)2]
 =
[
n(n+1)
2
]2
n(n)
 =
n2(n+1)2
n2×4
=
(n+1)2
4
 =
n2+2n+1
4
Hence, the sum to n terms is
Sn =
n
Σ
k=1
ak =
n
Σ
k=1
(
k2+2k+1
4
) =
1
4
[
n
Σ
k=1
k2+2
n
Σ
k=1
k+
n
Σ
k=1
1]
=
1
4
[
n(n+1)(2n+1)
6
+
2n(n+1)
2
+n]
=
n
4
[
(n+1)(2n+1)
6
+(n+1)+1]

=
n
4
[
2n2+n+2n+1+6n+6+6
6
]
=
n
24
(2n2+9n+13)
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