Question

Let $a\lt b\lt c$ 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 for $c?$

$-2$

$1$

$2$

$4$

$6$

Answer: C

Concepts:arithmetic sequence geometric sequence Diophantine Equation

Difficulty rating: 1630

Solution:

Let $d=b-a\gt0,$ so $b=a+d$ and $c=a+2d.$ Since $a,c,b$ is geometric, $\begin{gathered} \dfrac{c}{a}=\dfrac{b}{c}\\ \implies(a+2d)^2=a(a+d), \end{gathered}$ which simplifies to $3ad+4d^2=0,$ so $3a+4d=0.$

Then $a=-4k$ and $d=3k$ for a positive integer $k,$ giving $c=a+2d=2k.$ The smallest value is $c=2$ (with $a=-4,$ $b=-1,$ $c=2$ ).

Thus, the correct answer is C.

2014 AMC 12A Problem 14

Answer: C

Difficulty rating: 1630

Solution:

Problem 14 in Other Years