Difference between revisions of "2020 AMC 12B Problems/Problem 6"
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<cmath>\frac{(n+2)!-(n+1)!}{n!}</cmath> can be simplified by common denominator n!. | <cmath>\frac{(n+2)!-(n+1)!}{n!}</cmath> can be simplified by common denominator n!. | ||
Therefore, <cmath>\frac{(n+2)!-(n+1)!}{n!} = (n+2)(n+1)-(n+1)</cmath> | Therefore, <cmath>\frac{(n+2)!-(n+1)!}{n!} = (n+2)(n+1)-(n+1)</cmath> | ||
− | This expression can be shown as <cmath> | + | This expression can be shown as <cmath>(n+1)(n+2-1) = (n+1)^2</cmath> |
which proves that the answer is <math>\textbf{(D)}</math>. | which proves that the answer is <math>\textbf{(D)}</math>. |
Revision as of 20:34, 7 February 2020
Problem 6
For all integers the value of
is always which of the following?
Solution
can be simplified by common denominator n!.
Therefore,
This expression can be shown as
which proves that the answer is
.