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2011 AMC 12A Problems/Problem 8: Difference between revisions

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== Solution ==
== Solution ==
Let <math>A=x</math>. Then from <math>A+B+C=30</math>, we find that <math>B=25-x</math>. From <math>B+C+D=30</math>, we then get that <math>D=x</math>. Continuing this pattern, we find <math>E=25-x</math>, <math>F=5</math>, <math>G=x</math>, and finally <math>H=25-x</math>. So <math>A+H=x+25-x=25 \rightarrow \boxed{\textbf{C}}</math>
Let <math>A=x</math>. Then from <math>A+B+C=30</math>, we find that <math>B=25-x</math>. From <math>B+C+D=30</math>, we then get that <math>D=x</math>. Continuing this pattern, we find <math>E=25-x</math>, <math>F=5</math>, <math>G=x</math>, and finally <math>H=25-x</math>. So <math>A+H=x+25-x=25 \rightarrow \boxed{\textbf{C}}</math>
A faster technique is to assume that the problem can be solved, and thus <math>A+H</math> is an invariant.  Since <math>A + B + 5 = 30</math>, assign any value to <math>A</math>.  <math>10</math> is a simple value to plug in, which gives a value of <math>15</math> for B.  The 8-term sequence is thus <math>10, 15, 5, 10, 15, 5, 10, 15</math>.  The sum of the first and the last terms is <math>25\rightarrow \boxed{\textbf{C}}</math>
Note that this alternate solution is not a proof.  If the sum of <math>A+G</math> had been asked for, this technique would have given <math>20</math> as an answer, when the true answer would have been "cannot be determined".


== See also ==
== See also ==
{{AMC12 box|year=2011|num-b=7|num-a=9|ab=A}}
{{AMC12 box|year=2011|num-b=7|num-a=9|ab=A}}

Revision as of 19:15, 10 March 2011

Problem

In the eight term sequence $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A+H$?

$\textbf{(A)}\ 17 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 26 \qquad \textbf{(E)}\ 43$

Solution

Let $A=x$. Then from $A+B+C=30$, we find that $B=25-x$. From $B+C+D=30$, we then get that $D=x$. Continuing this pattern, we find $E=25-x$, $F=5$, $G=x$, and finally $H=25-x$. So $A+H=x+25-x=25 \rightarrow \boxed{\textbf{C}}$


A faster technique is to assume that the problem can be solved, and thus $A+H$ is an invariant. Since $A + B + 5 = 30$, assign any value to $A$. $10$ is a simple value to plug in, which gives a value of $15$ for B. The 8-term sequence is thus $10, 15, 5, 10, 15, 5, 10, 15$. The sum of the first and the last terms is $25\rightarrow \boxed{\textbf{C}}$

Note that this alternate solution is not a proof. If the sum of $A+G$ had been asked for, this technique would have given $20$ as an answer, when the true answer would have been "cannot be determined".

See also

2011 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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 12 Problems and Solutions
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