EX 11/02

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\[A = \frac{(n-1)!}{(n+1)!} = \frac{(n-1)!}{(n+1)n(n-1)!}=\frac 1{(n(n+1)}.\] \[\begin{aligned} B &= \frac{n!}{n} - (n-1)!& \\ &= \frac{n(n-1)!}{n} - (n-1)!& \\ &= (n-1)!-(n-1)! = 0.& \end{aligned}\] \[\begin{aligned} C &= \frac{(2n+1)!}{(2n-1)!}& \\ &= \frac{(2n+1)2n(2n-1)!}{(2n-1)!}& \\ &= (2n+1)2n& \\ &= 4n^2 + 2n.& \end{aligned}\] \[\begin{aligned} D &= \frac{(n-1)!}{n!} - \frac{n!}{(n+1)!}& \\ &=\frac{(n-1)!}{n(n-1)!} - \frac{n!}{(n+1)n!} = \frac 1{n} - \frac 1{n+1}& \\ &=\frac{n+1}{n(n+1)} - \frac{n}{n(n+1)} & \\ &= \frac{n+1-n}{n(n+1)}& \\ &= \frac 1{n(n+1)}.& \end{aligned}\]

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code : 3532