Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1986 AIME Problems/Problem 11"

(→‎See also)
m (cat, link, minor)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
−
The polynomial <math>1-x+x^2-x^3+\cdots+x^{16}-x^{17}</math> may be written in the form <math>a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}</math>, where <math>\displaystyle y=x+1</math> and thet <math>\displaystyle a_i</math>'s are constants. Find the value of <math>\displaystyle a_2</math>.
+
The [[polynomial]] <math>1-x+x^2-x^3+\cdots+x^{16}-x^{17}</math> may be written in the form <math>a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}</math>, where <math>\displaystyle y=x+1</math> and the <math>\displaystyle a_i</math>'s are [[constant]]s. Find the value of <math>\displaystyle a_2</math>.
 +
 
 
== Solution ==
 
== Solution ==
 
Since <math>(1+x)(1-x+x^2-x^3+\cdots +x^{16}-x^{17})=1-x^{18}</math>, we have
 
Since <math>(1+x)(1-x+x^2-x^3+\cdots +x^{16}-x^{17})=1-x^{18}</math>, we have
Line 6: Line 7:
 
<math>y(a_0+a_1y+a_2y^2+\cdots +a_{17}y^{17})=1-(y-1)^{18}</math>
 
<math>y(a_0+a_1y+a_2y^2+\cdots +a_{17}y^{17})=1-(y-1)^{18}</math>
  
 +
So <math>a_2</math> is the <math>y^3</math> [[coefficient]], which, by the [[Binomial Theorem]], is <math>\frac{18\cdot 17\cdot 16}{3\cdot 2\cdot 1}=3\cdot 17\cdot 16=816</math>
  
−
So <math>a_2</math> is the <math>y^3</math> coefficient, which, by the Binomial Theorem, is <math>\frac{18\cdot 17\cdot 16}{3\cdot 2\cdot 1}=3\cdot 17\cdot 16=816</math>
 
 
== See also ==
 
== See also ==
 
{{AIME box|year=1986|num-b=10|num-a=12}}
 
{{AIME box|year=1986|num-b=10|num-a=12}}
Line 13: Line 14:
 
* [[American Invitational Mathematics Examination]]
 
* [[American Invitational Mathematics Examination]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
 +
 +
[[Category:Intermediate Algebra Problems]]

Revision as of 18:21, 12 September 2007

Problem

The polynomial $1-x+x^2-x^3+\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $\displaystyle y=x+1$ and the $\displaystyle a_i$'s are constants. Find the value of $\displaystyle a_2$.

Solution

Since $(1+x)(1-x+x^2-x^3+\cdots +x^{16}-x^{17})=1-x^{18}$, we have

$y(a_0+a_1y+a_2y^2+\cdots +a_{17}y^{17})=1-(y-1)^{18}$

So $a_2$ is the $y^3$ coefficient, which, by the Binomial Theorem, is $\frac{18\cdot 17\cdot 16}{3\cdot 2\cdot 1}=3\cdot 17\cdot 16=816$

See also

1986 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1986_AIME_Problems/Problem_11&oldid=16078"