Difference between revisions of "1993 AIME Problems/Problem 5"
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== Problem == | == Problem == | ||
| − | Let <math>P_0(x) = x^3 + 313x^2 - 77x - 8\,</math>. For [[integer]]s <math>n \ge 1\,</math>, define <math>P_n(x) = P_{n - 1}(x - n)\,</math>. What is the [[coefficient]] of <math>x\,</math> in <math>P_{20}(x)\,</math>? | + | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>Let <math>P_0(x) = x^3 + 313x^2 - 77x - 8\,</math>. For [[integer]]s <math>n \ge 1\,</math>, define <math>P_n(x) = P_{n - 1}(x - n)\,</math>. What is the [[coefficient]] of <math>x\,</math> in <math>P_{20}(x)\,</math>?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> |
== Solution == | == Solution == | ||
Revision as of 18:15, 11 November 2015
Problem
Let 
. For integers 
, define 
. What is the coefficient of 
in 
?
Solution
Notice that
Using the formula for the sum of the first 
numbers, 
. Therefore, ![\[P_{20}(x) = P_0(x - 210).\]](http://latex.artofproblemsolving.com/6/3/f/63f8ce4cbf48964dd61df68155e7c6b185964047.png)
Substituting 
into the function definition, we get 
. We only need the coefficients of the linear terms, which we can find by the binomial theorem.
will have a linear term of 
.
will have a linear term of 
.
will have a linear term of 
.
will have a linear term of 
.
will have a linear term of 
.
will have a linear term of 
.Adding up the coefficients, we get 
.
See also
| 1993 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
