Difference between revisions of "2020 AMC 12B Problems/Problem 6"
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==Solution== | ==Solution== | ||
− | <cmath>\frac{(n+2)!-(n+1)!}{n!} | + | We first expand the expression: |
− | + | <cmath>\frac{(n+2)!-(n+1)!}{n!} = \frac{(n+2)(n+1)n!-(n+1)n!}{n!}</cmath> | |
− | + | ||
− | which proves that the answer is <math>\textbf{(D)}</math>. | + | We can now divide out a common factor of <math>n!</math> from each term of this expression: |
+ | |||
+ | <cmath>(n+2)(n+1)-(n+1)</cmath> | ||
+ | |||
+ | Factoring out <math>(n+1)</math>, we get <cmath>(n+1)(n+2-1) = (n+1)^2</cmath> | ||
+ | |||
+ | which proves that the answer is <math>\boxed{\textbf{(D) a perfect square}}</math>. |
Revision as of 19:35, 7 February 2020
Problem 6
For all integers the value of is always which of the following?
Solution
We first expand the expression:
We can now divide out a common factor of from each term of this expression:
Factoring out , we get
which proves that the answer is .