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 "1969 AHSME Problems/Problem 32"

(Created page with "== Problem == Let a sequence <math>\{u_n\}</math> be defined by <math>u_1=5</math> and the relationship <math>u_{n+1}-u_n=3+4(n-1), n=1,2,3\cdots.</math>If <math>u_n</math> is e...")
 
(Solution to Problem 32)
Line 10: Line 10:
  
 
== Solution ==
 
== Solution ==
−
<math>\fbox{C}</math>
+
 
 +
Note that the first differences create a linear function, so the sequence <math>{u_n}</math> is quadratic.
 +
 
 +
The first three terms of the sequence are <math>5</math>, <math>12</math>, and <math>23</math>.  From there, a [[system of equations]] can be written.
 +
<cmath>a+b+c=5</cmath>
 +
<cmath>4a+2b+c=12</cmath>
 +
<cmath>9a+3b+c=23</cmath>
 +
Solve the system to get <math>a=2</math>, <math>b=1</math>, and <math>c=2</math>.  The sum of the coefficients is <math>\boxed{\textbf{(C) } 5}</math>.
  
 
== See also ==
 
== See also ==

Revision as of 03:06, 10 June 2018

Problem

Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relationship $u_{n+1}-u_n=3+4(n-1), n=1,2,3\cdots.$If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is:

$\text{(A) 3} \quad \text{(B) 4} \quad \text{(C) 5} \quad \text{(D) 6} \quad \text{(E) 11}$

Solution

Note that the first differences create a linear function, so the sequence ${u_n}$ is quadratic.

The first three terms of the sequence are $5$, $12$, and $23$. From there, a system of equations can be written. \[a+b+c=5\] \[4a+2b+c=12\] \[9a+3b+c=23\] Solve the system to get $a=2$, $b=1$, and $c=2$. The sum of the coefficients is $\boxed{\textbf{(C) } 5}$.

See also

1969 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 31 		Followed by
Problem 33
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1969_AHSME_Problems/Problem_32&oldid=95029"