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Mathematics
Mathematics
Problem
The two limits
lim
n
→
∞
∑
i
=
1
n
(
A
(
2
+
2
i
n
)
+
B
)
⋅
2
n
=
30
and
lim
n
→
∞
∑
i
=
1
n
(
A
(
2
+
i
n
)
+
B
)
⋅
1
n
=
12
\lim_{n\to\infty}\sum_{i=1}^{n}\bigl(A\bigl(2+\tfrac{2i}{n}\bigr)+B\bigr)\cdot\tfrac{2}{n}=30\quad\text{and}\quad\lim_{n\to\infty}\sum_{i=1}^{n}\bigl(A\bigl(2+\tfrac{i}{n}\bigr)+B\bigr)\cdot\tfrac{1}{n}=12
n
→
∞
lim
i
=
1
∑
n
(
A
(
2
+
n
2
i
)
+
B
)
⋅
n
2
=
30
and
n
→
∞
lim
i
=
1
∑
n
(
A
(
2
+
n
i
)
+
B
)
⋅
n
1
=
12
are both finite. Determine the ordered pair
(
A
,
B
)
(A,B)
(
A
,
B
)
.
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