2020 AMC 12B Problems/Problem 6
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Problem 6
For all integers 
the value of
![\[\frac{(n+2)!-(n+1)!}{n!}\]](http://latex.artofproblemsolving.com/e/4/c/e4cf3e216ff29ec12bb85e389993280b1ca40180.png)
is always which of the following?
Solution
can be simplified by common denominator n!.
Therefore, ![\[\frac{(n+2)!-(n+1)!}{n!} = (n+2)(n+1)-(n+1)\]](http://latex.artofproblemsolving.com/c/2/2/c2267385bac56e31aba19576629bceb9f07127e6.png)
This expression can be shown as
\[\(n+1)(n+2-1) = (n+1)^2\] (Error compiling LaTeX. Unknown error_msg)
which proves that the answer is $\textbf{(D)}.
