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

Revision as of 15:40, 22 December 2015

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 $\frac{a}{d} = \frac{a}{a + 3n} = \frac{a}{4a} = \boxed{\frac{1}{4}} \Longrightarrow \mathrm{(C)}$ .

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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.

@@ Line 25: / Line 25: @@
 <cmath>\frac{a + n}{a} = \frac{a + 3n}{a + n}.</cmath>
 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 = a</cmath>

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