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Problem
Problem
For positive integers
K
=
1
,
2
,
3
,
…
n
K = 1, 2, 3, \dots n
K
=
1
,
2
,
3
,
…
n
. Let
S
k
S_k
S
k
denotes the area of
△
O
A
B
K
\triangle OAB_K
△
O
A
B
K
(where 'O' is the origin) such that
∠
A
O
B
K
=
K
π
2
n
\angle AOB_K = \frac{K\pi}{2n}
∠
A
O
B
K
=
2
n
K
π
,
O
A
=
1
OA = 1
O
A
=
1
and
O
B
K
=
K
OB_K = K
O
B
K
=
K
. The value of the
lim
n
→
∞
1
n
2
⋅
∑
K
=
1
n
S
K
\lim_{n \to \infty} \frac{1}{n^2} \cdot \sum_{K=1}^n S_K
lim
n
→
∞
n
2
1
⋅
∑
K
=
1
n
S
K
is a.
2
π
2
\frac{2}{\pi^2}
π
2
2
b.
4
π
2
\frac{4}{\pi^2}
π
2
4
c.
8
π
2
\frac{8}{\pi^2}
π
2
8
d.
1
2
π
2
\frac{1}{2\pi^2}
2
π
2
1
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