Difference between revisions of "2008 AMC 10B Problems/Problem 11"

(New page: ==Problem== {{problem}} ==Solution== {{solution}} ==See also== {{AMC10 box|year=2008|ab=B|num-b=10|num-a=12}})
 
(Solution)
(9 intermediate revisions by 6 users not shown)
Line 1: Line 1:
 
==Problem==
 
==Problem==
{{problem}}
+
Suppose that <math>(u_n)</math> is a [[sequence]] of real numbers satifying <math>u_{n+2}=2u_{n+1}+u_n</math>,
 +
 
 +
and that <math>u_3=9</math> and <math>u_6=128</math>. What is <math>u_5</math>?
 +
 
 +
<math>\mathrm{(A)}\ 40\qquad\mathrm{(B)}\ 53\qquad\mathrm{(C)}\ 68\qquad\mathrm{(D)}\ 88\qquad\mathrm{(E)}\ 104</math>
  
 
==Solution==
 
==Solution==
{{solution}}
+
Plugging in <math>n=4</math>, we get
 +
 
 +
<center><math>128=2u_5+u_4.</math></center>
 +
 
 +
Plugging in <math>n=3</math>, we get
 +
 
 +
<center><math>u_5=2u_4+9.</math></center>
 +
 
 +
This is simply a system of two equations with two unknowns. Substituting gives <math>128=5u_4+18 \Longrightarrow u_4=22</math>, and <math>u_5=\frac{128-22}{2}=53 \longleftarrow \textbf{(B)}</math>.
  
 
==See also==
 
==See also==
 
{{AMC10 box|year=2008|ab=B|num-b=10|num-a=12}}
 
{{AMC10 box|year=2008|ab=B|num-b=10|num-a=12}}
 +
 +
[[Category:Introductory Algebra Problems]]
 +
{{MAA Notice}}

Revision as of 21:45, 13 November 2017

Problem

Suppose that $(u_n)$ is a sequence of real numbers satifying $u_{n+2}=2u_{n+1}+u_n$,

and that $u_3=9$ and $u_6=128$. What is $u_5$?

$\mathrm{(A)}\ 40\qquad\mathrm{(B)}\ 53\qquad\mathrm{(C)}\ 68\qquad\mathrm{(D)}\ 88\qquad\mathrm{(E)}\ 104$

Solution

Plugging in $n=4$, we get

$128=2u_5+u_4.$

Plugging in $n=3$, we get

$u_5=2u_4+9.$

This is simply a system of two equations with two unknowns. Substituting gives $128=5u_4+18 \Longrightarrow u_4=22$, and $u_5=\frac{128-22}{2}=53 \longleftarrow \textbf{(B)}$.

See also

2008 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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
All AMC 10 Problems and Solutions

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

Invalid username
Login to AoPS