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Difference between revisions of "2013 AMC 10B Problems/Problem 21"
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Let the first two terms of the first sequence be <math>x_{1}</math> and <math>x_{2}</math> and the first two of the second sequence be <math>y_{1}</math> and <math>y_{2}</math>.  Computing the seventh term, we see that <math>5x_{1} + 8x_{2} = 5y_{1} + 8y_{2}</math>.  Note that this means that <math>x_{1}</math> and <math>x_{2}</math> must have the same value modulo 8.  To minimize, let one of them be 0; WLOG assume that <math>x_{1} = 0</math>.  Thus, the smallest possible value of <math>y_{1}</math> is <math>8</math>; since the sequences are nondecreasing <math>y_{2} \ge 8</math>.  To minimize, let <math>y_{2} = 8</math>.  Thus, <math>5y_{1} + 8y_{2} = 40 + 64 = \boxed{\textbf{(C) }104}</math>.
 
Let the first two terms of the first sequence be <math>x_{1}</math> and <math>x_{2}</math> and the first two of the second sequence be <math>y_{1}</math> and <math>y_{2}</math>.  Computing the seventh term, we see that <math>5x_{1} + 8x_{2} = 5y_{1} + 8y_{2}</math>.  Note that this means that <math>x_{1}</math> and <math>x_{2}</math> must have the same value modulo 8.  To minimize, let one of them be 0; WLOG assume that <math>x_{1} = 0</math>.  Thus, the smallest possible value of <math>y_{1}</math> is <math>8</math>; since the sequences are nondecreasing <math>y_{2} \ge 8</math>.  To minimize, let <math>y_{2} = 8</math>.  Thus, <math>5y_{1} + 8y_{2} = 40 + 64 = \boxed{\textbf{(C) }104}</math>.
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== See also ==
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{{AMC10 box|year=2013|ab=B|num-b=20|num-a=22}}

Revision as of 16:05, 27 March 2013

Problem

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. What is the smallest possible value of N?

$\textbf{(A)}\ 55 \qquad\textbf{(B)}\ 89 \qquad\textbf{(C)}\ 104 \qquad\textbf{(D)}\ 144 \qquad\textbf{(E)}\ 273$

Solution

Let the first two terms of the first sequence be $x_{1}$ and $x_{2}$ and the first two of the second sequence be $y_{1}$ and $y_{2}$. Computing the seventh term, we see that $5x_{1} + 8x_{2} = 5y_{1} + 8y_{2}$. Note that this means that $x_{1}$ and $x_{2}$ must have the same value modulo 8. To minimize, let one of them be 0; WLOG assume that $x_{1} = 0$. Thus, the smallest possible value of $y_{1}$ is $8$; since the sequences are nondecreasing $y_{2} \ge 8$. To minimize, let $y_{2} = 8$. Thus, $5y_{1} + 8y_{2} = 40 + 64 = \boxed{\textbf{(C) }104}$.

See also

2013 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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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