Difference between revisions of "2014 AMC 12A Problems/Problem 14"

Latest revision as of 16:25, 19 September 2022

Problem

Let $a<b<c$ be three integers such that $a,b,c$ is an arithmetic progression and $a,c,b$ is a geometric progression. What is the smallest possible value of $c$ ?

$\textbf{(A) }-2\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }4\qquad \textbf{(E) }6\qquad$

Solution 1

We have $b-a=c-b$ , so $a=2b-c$ . Since $a,c,b$ is geometric, $c^2=ab=(2b-c)b \Rightarrow 2b^2-bc-c^2=(2b+c)(b-c)=0$ . Since $a<b<c$ , we can't have $b=c$ and thus $c=-2b$ . Then our arithmetic progression is $4b,b,-2b$ . Since $4b < b < -2b$ , $b < 0$ . The smallest possible value of $c=-2b$ is $(-2)(-1)=2$ , or $\boxed{\textbf{(C)}}$ .

(Solution by AT)

Solution 2

Taking the definition of an arithmetic progression, there must be a common difference between the terms, giving us $(b-a) = (c-b)$ . From this, we can obtain the expression $a = 2b-c$ . Again, by taking the definition of a geometric progression, we can obtain the expression, $c=ar$ and $b=ar^2$ , where r serves as a value for the ratio between two terms in the progression. By substituting $b$ and $c$ in the arithmetic progression expression with the obtained values from the geometric progression, we obtain the equation, $a=2ar^2-ar$ which can be simplified to $(r-1)(2r+1)=0$ giving us $r=1$ or $r=-1/2$ . Thus, from the geometric progression, $a=a$ , $b=1/4a$ and $c=-1/2a$ . Looking at the initial conditions of $a<b<c$ we can see that the lowest integer value that would satisfy the above expressions is if $a = -4$ , thus making $c=2$ or $\boxed{\textbf{(C)}}$ (Solution by thatuser)

Solution 3

By the definition of an arithmetic progression, we can label the terms $a$ , $b$ , and $c$ , as $a$ , $a+d$ , and $a+2d$ . Now, we have that $a$ , $a+2d$ , and $a+d$ form a geometric progression. Since a geometric ratio has a common ratio between terms, we have $(a+2d)/a = (a+d)/a+2d$ . Cross multiplying, we obtain the equation $(a+2d)^2=a(a+d)$ . Multiplying it out and cancelling terms, we are left with the quadratic equation $4d^2+3ad=0$ . Solving for $d$ in terms of $a$ , we get that $d=-3a/4$ or $d=0$ . Looking at the problem, we know that the $d$ cannot be 0, therefore the arithmetic progression is $a, a/4, -a/2$ , so we need to find the minimum value of $-a/2$ while $-a/2>a$ . Looking at our progression, we realize that a must be a multiple of 4 so that every term is an integer. Substituting $a=-4$ , since that would yield the smallest value of $-a/2$ which satisfies the conditions, we figure out that the answer is $\boxed{\textbf{(C)}}$ . (Solution by i8Pie)

Video Solution

https://youtu.be/sxBuSZGfmHg

~Lucas

@@ Line 9: / Line 9: @@
 \textbf{(E) }6\qquad</math>
-==Solution==
+==Solution 1==
 We have <math>b-a=c-b</math>, so <math>a=2b-c</math>. Since <math>a,c,b</math> is geometric, <math>c^2=ab=(2b-c)b \Rightarrow 2b^2-bc-c^2=(2b+c)(b-c)=0</math>. Since <math>a<b<c</math>, we can't have <math>b=c</math> and thus <math>c=-2b</math>. Then our arithmetic progression is <math>4b,b,-2b</math>. Since <math>4b < b < -2b</math>, <math>b < 0</math>. The smallest possible value of <math>c=-2b</math> is <math>(-2)(-1)=2</math>, or <math>\boxed{\textbf{(C)}}</math>.
-(Solution by AwesomeToad)
+(Solution by AT)
+==Solution 2==
+Taking the definition of an arithmetic progression, there must be a common difference between the terms, giving us <math>(b-a) = (c-b)</math>. From this, we can obtain the expression <math>a = 2b-c</math>. Again, by taking the definition of a geometric progression, we can obtain the expression, <math>c=ar</math> and <math>b=ar^2</math>, where r serves as a value for the ratio between two terms in the progression. By substituting <math>b</math> and <math>c</math> in the arithmetic progression expression with the obtained values from the geometric progression, we obtain the equation, <math>a=2ar^2-ar</math> which can be simplified to <math>(r-1)(2r+1)=0</math> giving us <math>r=1</math> or <math>r=-1/2</math>. Thus, from the geometric progression, <math>a=a</math>, <math>b=1/4a</math> and <math>c=-1/2a</math>. Looking at the initial conditions of <math>a<b<c</math> we can see that the lowest integer value that would satisfy the above expressions is if <math>a = -4</math>, thus making <math>c=2</math> or <math>\boxed{\textbf{(C)}}</math>
+(Solution by thatuser)
+==Solution 3==
+By the definition of an arithmetic progression, we can label the terms <math>a</math>, <math>b</math>, and <math>c</math>, as <math>a</math>, <math>a+d</math>, and <math>a+2d</math>. Now, we have that <math>a</math>, <math>a+2d</math>, and <math>a+d</math> form a geometric progression. Since a geometric ratio has a common ratio between terms, we have <math>(a+2d)/a = (a+d)/a+2d</math>. Cross multiplying, we obtain the equation <math>(a+2d)^2=a(a+d)</math>. Multiplying it out and cancelling terms, we are left with the quadratic equation <math>4d^2+3ad=0</math>. Solving for <math>d</math> in terms of <math>a</math>, we get that <math>d=-3a/4</math> or <math>d=0</math>. Looking at the problem, we know that the <math>d</math> cannot be 0, therefore the arithmetic progression is <math>a, a/4, -a/2</math>, so we need to find the minimum value of <math>-a/2</math> while <math>-a/2>a</math>. Looking at our progression, we realize that a must be a multiple of 4 so that every term is an integer. Substituting <math>a=-4</math>, since that would yield the smallest value of <math>-a/2</math> which satisfies the conditions, we figure out that the answer is <math>\boxed{\textbf{(C)}}</math>.
+(Solution by i8Pie)
+==Video Solution==
+https://youtu.be/sxBuSZGfmHg
+~Lucas
 ==See Also==
 {{AMC12 box|year=2014|ab=A|num-b=13|num-a=15}}
 {{MAA Notice}}

2014 AMC 12A (Problems • Answer Key • Resources)
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