AMC 10 · 2003 · #6

Grade 8 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
sequences-geometricexponentssigned-square-root work-backwards ↑ Prerequisites: sequences-geometric
📏 Medium solution 💡 2 insights
Problem
In a geometric sequence every term is the one before it multiplied by the same fixed number. The second term is 2 and the fourth term is 6. Find which of the five listed values could be the first term.

Pick an answer.

(A)
$-\sqrt{3}$
(B)
$-\frac{2\sqrt{3}}{3}$
(C)
$-\frac{\sqrt{3}}{3}$
(D)
$\sqrt{3}$
(E)
3
How to solve
Strategy Introduce a Variable

The first term is unknown and so is the ratio, so Tool #4 (Introduce a Variable) is the spine: call the first term a and the ratio r, then the two facts become ar = 2 and ar³ = 6. Tool #11 (Work Backwards) does the heavy lifting — dividing the fourth-term equation by the second-term equation cancels a and leaves a clean equation for r, and once r is known one step back from the second term recovers a. Tool #3 (Eliminate Possibilities) finishes it: the two facts allow two ratios (a positive and a negative one), giving two candidate first terms, and only one of those candidates appears among the five choices.

1STEP 1

Name the unknowns, write the facts

Naming the first term and ratio writes the two facts as a r = 2 and a r³ = 6.

a r = 2 a r³ = 6
2STEP 2

Divide to cancel the first term

Dividing cancels the first term and leaves r² = 3.

(a r³)/(a r) = 6/2 → r² = 3
3STEP 3

Take the square root — keep both signs

The square root has two signs, so both ratios must be carried.

r² = 3 → r = √(3) or r = -√(3)
4STEP 4

Step back to the first term

Stepping back and rationalising gives first terms plus or minus 2√(3)/3.

a = 2/±√(3) = ±2/√(3)·√(3)/√(3) = ±2√(3)/3
5STEP 5

Match a candidate to the choices

Only the negative one is listed, so the answer is -2√(3)/3, choice (B).

a ∈ { 2√(3)/3, -2√(3)/3 } → -2√(3)/3 = (B)
Answer
-2√(3)/3
Test the winner directly. If a = -2√(3)/3, then for the second term to be 2 the ratio must be r = 2/a = 2/(-2√(3)/3) = -√(3). Building the sequence: first -2√(3)/3, second -2√(3)/3·(-√(3)) = (2 · 3)/3 = 2 (correct), third 2·(-√(3)) = -2√(3), fourth -2√(3)·(-√(3)) = 2 · 3 = 6 (correct). The second and fourth terms come out to 2 and 6, so choice (B) genuinely works.
💡Key takeaway

Name the first term and the ratio, divide the fourth-term equation by the second to get the ratio squared, and remember the square root has two signs.

  • Name the unknowns, write the facts
  • Divide to cancel the first term
  • Take the square root — keep both signs
  • Step back to the first term
  • Match a candidate to the choices

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