AMC 10 · 2002 · #9

Grade 9 algebra
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up
sequences-arithmeticsequences-geometricquadratic-equationsratio-proportion convert-to-algebra ↑ Prerequisites: sequences-arithmeticsequences-geometric
📏 Medium solution 💡 2 insights
Problem
Four positive numbers a, b, c, d are given, and they satisfy two pattern conditions at once. Read all four in order, they form an increasing arithmetic sequence: the step from each to the next is the same, and that step is positive. Read only a, b, d — skipping c entirely — they form a geometric sequence: each is the one before it multiplied by a fixed ratio. Find the value of a/d.

Pick an answer.

(A)
$\frac 1{12}$
(B)
$\frac 16$
(C)
$\frac 14$
(D)
$\frac 13$
(E)
$\frac 12$
How to solve
Strategy Introduce a Variable

Two pattern words are attached to overlapping lists, so the job is to write both patterns in the same letters and let them collide. Tool #4 (Introduce a Variable) does that: one starting value a and one common difference n describe all four numbers, so the arithmetic condition is used up completely and nothing is left vague. Tool #13 (Convert to Algebra) turns the word 'geometric' into the single equation b² = ad, which is the only real equation in the problem. Substituting one into the other produces a quadratic whose two roots must both be examined — and this is the step where a careless solver goes wrong, because the root that gets thrown away is thrown away for being non-increasing, not for being negative. Tool #15 (Organize Information in More Ways) finishes by rewriting all four numbers as multiples of a, which shows the requested ratio at a glance and shows that a itself is genuinely free.

1STEP 1

Write all four with one step size

One step size n writes all four, with n greater than zero; note c never reappears.

b = a+n, c = a+2n, d = a+3n, n ＞ 0
2STEP 2

Turn 'geometric' into one equation

Equal ratios cross-multiply into the single equation b² = ad.

b/a = d/b ⇔ b² = ad
3STEP 3

Substitute, and read both roots

Substituting collapses to n(n-a)=0; the root n=0 breaks the increasing rule, so n = a.

(a+n)² = a(a+3n) → a²+2an+n² = a²+3an → n² - an = 0 → n(n-a) = 0 → n = 0 or n = a
4STEP 4

Rewrite the four numbers and read off the ratio

The four become a, 2a, 3a, 4a, so the ratio is 1/4 regardless of a, choice (C).

n = a → (a,b,c,d) = (a, 2a, 3a, 4a); a/d = a/4a = 1/4 → (C)
Answer
1/4
Test the smallest case directly: 1, 2, 3, 4. The differences are 1, 1, 1, positive and equal, so it is an increasing arithmetic sequence; and 1, 2, 4 doubles each time, so it is geometric. Then a/d = 1/4. Testing one case is only a check, not a proof — but step 3 supplies the proof, because n(n-a)=0 forced n = a with no other option, so no differently shaped quadruple exists. The wrong choices line up in a revealing way: a/b = 1/2 is (E) and a/c = 1/3 is (D), so choices (D) and (E) are precisely the answers of someone who stopped at the wrong term — a good reason to reread which letter the question asked about. And 1/4 passing the sanity test a/d ＜ 1 is expected, since the sequence is increasing so d ＞ a.
💡Key takeaway

When two pattern words describe overlapping lists, write both patterns in the same letters — the equation that results usually has two roots, and it is the wording of the problem, not the sign, that tells you which root to throw away.

  • Write all four with one step size
  • Turn 'geometric' into one equation
  • Substitute, and read both roots
  • Rewrite the four numbers and read off the ratio

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