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Mathematics
Problem
Which of the following expressions equals the right‐endpoint Riemann sum with
n
n
n
equal subintervals that approximates
∫
2
5
1
x
d
x
\displaystyle \int_{2}^{5} \frac{1}{x}\,dx
∫
2
5
x
1
d
x
? a)
∑
k
=
0
n
−
1
1
2
+
3
k
n
⋅
3
n
\displaystyle \sum_{k=0}^{n-1} \frac{1}{2+\dfrac{3k}{n}}\cdot\frac{3}{n}
k
=
0
∑
n
−
1
2
+
n
3
k
1
⋅
n
3
b)
∑
k
=
1
n
1
2
+
3
k
n
⋅
3
n
\displaystyle \sum_{k=1}^{n} \frac{1}{2+\dfrac{3k}{n}}\cdot\frac{3}{n}
k
=
1
∑
n
2
+
n
3
k
1
⋅
n
3
c)
∑
k
=
1
n
1
2
+
3
(
k
−
0.5
)
n
⋅
3
n
\displaystyle \sum_{k=1}^{n} \frac{1}{2+\dfrac{3(k-0.5)}{n}}\cdot\frac{3}{n}
k
=
1
∑
n
2
+
n
3
(
k
−
0.5
)
1
⋅
n
3
d)
∑
k
=
1
n
1
2
+
3
k
n
⋅
n
3
\displaystyle \sum_{k=1}^{n} \frac{1}{2+\dfrac{3k}{n}}\cdot\frac{n}{3}
k
=
1
∑
n
2
+
n
3
k
1
⋅
3
n
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