Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2002 AMC 12B Problems/Problem 9"

(added new solution)
m (→‎Solution 1)
 
(2 intermediate revisions by 2 users not shown)
Line 10: Line 10:
 
==Solution==
 
==Solution==
 
=== Solution 1 ===
 
=== Solution 1 ===
−
We can let a=1, b=2, c=3, and d=4. <math>\frac{a}{d}=\boxed{\frac{1}{4}}  \Longrightarrow \mathrm{(C)}</math>
+
We can let <math>a=1</math>, <math>b=2</math>, <math>c=3</math>, and <math>d=4</math>. <math>\frac{a}{d}=\boxed{\frac{1}{4}}  \Longrightarrow \mathrm{(C)}</math>
  
 
=== Solution 2 ===
 
=== Solution 2 ===
Line 25: Line 25:
 
<cmath>\frac{a + n}{a} = \frac{a + 3n}{a + n}.</cmath>
 
<cmath>\frac{a + n}{a} = \frac{a + 3n}{a + n}.</cmath>
 
Cross-multiplying, we get  
 
Cross-multiplying, we get  
−
<cmath>a^2 + 2an + n^2 &= a^2 + 3an</cmath>
+
<cmath>a^2 + 2an + n^2 = a^2 + 3an</cmath>
 
<cmath>n^2 = an</cmath>
 
<cmath>n^2 = an</cmath>
 
<cmath>n = a</cmath>
 
<cmath>n = a</cmath>
Line 34: Line 34:
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 20:42, 11 December 2017

Problem

If $a,b,c,d$ are positive real numbers such that $a,b,c,d$ form an increasing arithmetic sequence and $a,b,d$ form a geometric sequence, then $\frac ad$ is

$\mathrm{(A)}\ \frac 1{12} \qquad\mathrm{(B)}\ \frac 16 \qquad\mathrm{(C)}\ \frac 14 \qquad\mathrm{(D)}\ \frac 13 \qquad\mathrm{(E)}\ \frac 12$

Solution

Solution 1

We can let $a=1$, $b=2$, $c=3$, and $d=4$. $\frac{a}{d}=\boxed{\frac{1}{4}} \Longrightarrow \mathrm{(C)}$

Solution 2

As $a, b, d$ is a geometric sequence, let $b=ka$ and $d=k^2a$ for some $k>0$.

Now, $a, b, c, d$ is an arithmetic sequence. Its difference is $b-a=(k-1)a$. Thus $d=a + 3(k-1)a = (3k-2)a$.

Comparing the two expressions for $d$ we get $k^2=3k-2$. The positive solution is $k=2$, and $\frac{a}{d}=\frac{a}{k^2a}=\frac{1}{k^2}=\boxed{\frac{1}{4}} \Longrightarrow \mathrm{(C)}$.

Solution 3

Letting $n$ be the common difference of the arithmetic progression, we have $b = a + n$, $c = a + 2n$, $d = a + 3n$. We are given that $b / a$ = $d / b$, or \[\frac{a + n}{a} = \frac{a + 3n}{a + n}.\] Cross-multiplying, we get \[a^2 + 2an + n^2 = a^2 + 3an\] \[n^2 = an\] \[n = a\] So $\frac{a}{d} = \frac{a}{a + 3n} = \frac{a}{4a} = \boxed{\frac{1}{4}} \Longrightarrow \mathrm{(C)}$.

See also

2002 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2002_AMC_12B_Problems/Problem_9&oldid=88872"