Difference between revisions of "2008 AMC 10B Problems/Problem 11"
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==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>? | ||
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+ | <math>\mathrm{(A)}\ 40\qquad\mathrm{(B)}\ 53\qquad\mathrm{(C)}\ 68\qquad\mathrm{(D)}\ 88\qquad\mathrm{(E)}\ 104</math> | ||
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==Solution== | ==Solution== | ||
− | + | Plugging in <math>n=4</math>, we get | |
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− | + | <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 | + | This is a system of two equations with two unknowns. Multiplying the second equation by 2 and substituting into the first equation gives <math>128=5u_4+18 \Longrightarrow u_4=22</math>, and <math>u_5=\frac{128-22}{2}=53 \longrightarrow \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}} | ||
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+ | [[Category:Introductory Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 12:40, 7 June 2021
Problem
Suppose that is a sequence of real numbers satifying ,
and that and . What is ?
Solution
Plugging in , we get
Plugging in , we get
This is a system of two equations with two unknowns. Multiplying the second equation by 2 and substituting into the first equation gives , and .
See also
2008 AMC 10B (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 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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