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Mathematics
Mathematics
Problem
The constants
m
m
m
and
b
b
b
satisfy both of the following equalities:
lim
n
→
∞
∑
i
=
1
n
(
m
(
4
i
n
)
+
b
)
4
n
=
40
\lim_{n\to\infty}\sum_{i=1}^{n}\bigl(m\left(\tfrac{4i}{n}\right)+b\bigr)\frac{4}{n}=40
n
→
∞
lim
i
=
1
∑
n
(
m
(
n
4
i
)
+
b
)
n
4
=
40
and
lim
n
→
∞
∑
i
=
1
n
(
m
(
2
i
n
)
+
b
)
2
n
=
12.
\lim_{n\to\infty}\sum_{i=1}^{n}\bigl(m\left(\tfrac{2i}{n}\right)+b\bigr)\frac{2}{n}=12.
n
→
∞
lim
i
=
1
∑
n
(
m
(
n
2
i
)
+
b
)
n
2
=
12.
Which ordered pair
(
m
,
b
)
(m,b)
(
m
,
b
)
satisfies these conditions? a)
(
3
,
3
)
(3,3)
(
3
,
3
)
b)
(
4
,
−
2
)
(4,-2)
(
4
,
−
2
)
c)
(
4
,
2
)
(4,2)
(
4
,
2
)
d)
(
2
,
2
)
(2,2)
(
2
,
2
)
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