Problem: When simplified the product (1−13)(1−14)(1−15)…(1−1n)(1-\frac{1}{3})(1- \frac{1}{4})(1- \frac{1}{5}) \ldots(1- \frac{1}{n})(1−31)(1−41)(1−51)…(1−n1) becomes:
Answer Choices:
A. 1/n1 / n1/n
B. 2/n2 / n2/n
C. 2(n−1)n\dfrac{2(n-1)}{n}n2(n−1)
D. 2n(n+1)\dfrac{2}{n(n+1)}n(n+1)2
E. 3n(n+1)\dfrac{3}{n(n+1)}n(n+1)3 Solution:
(1−13)(1−14)(1−15)…(1−1n)\left(1-\dfrac{1}{3}\right) \left(1-\dfrac{1}{4}\right)\left(1-\dfrac{1}{5}\right) \ldots\left(1-\dfrac{1}{n}\right) (1−31)(1−41)(1−51)…(1−n1)
=23⋅34⋅48⋅⋯n−2n−1n−1n=2n=\dfrac{2}{3} \cdot \dfrac{3}{4} \cdot \dfrac{4}{8} \cdot \cdots \dfrac{n-2}{n-1} \dfrac{n-1}{n}=\dfrac{2}{n} =32⋅43⋅84⋅⋯n−1n−2nn−1=n2