2002 AMC 12B Problems/Problem 9

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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

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

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 $\frac{a}{d}=\frac{a}{k^2a}=\frac{1}{k^2}=\boxed{\frac{1}{4}}$ .

2002 AMC 12B (Problems • 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

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