AMC 10 · 2004 · #18
Grade 8 algebraPick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Name the common difference d (Tool #4): then the arithmetic terms are 9, 9+d, 9+2d, and after the additions the geometric terms are 9, 11+d, 29+2d. The geometric condition — middle term squared equals the product of the ends — turns the whole situation into a single equation in d (Tool #13), which is a quadratic. A quadratic can have two solutions, so there can be two valid common differences and therefore two possible third terms. The word "smallest" is the signal to solve completely and then take the minimum (Tool #14, Extreme Principle) — the trap is stopping at the first (positive-difference) answer 49 and missing the smaller one that a negative difference produces.
Name the common difference
Let d be the common difference: the arithmetic terms are 9, 9+d, 9+2d, so the geometric terms become 9, 11+d, and 29+2d.
One unknown, the common difference, controls all three terms, so naming it captures the entire problem.
7.EE.B.4Introduce A VariableUse the geometric-progression rule
Equal ratios mean middle squared equals the ends multiplied: (11+d)² = 9(29+2d), which expands to d² + 4d - 140 = 0.
Equal ratios cross-multiply into 'middle squared equals ends multiplied,' which converts the pattern into one solvable equation.
Equal ratios cross-multiply into the middle squared equalling the ends multiplied.
▸ Why?
A geometric run multiplies by the same number each step, so the two neighbouring ratios must agree.
▸ Why?
The three terms were first written from one start and one fixed step, which is what leaves a single unknown.
Solve for the common difference
Solving d² + 4d - 140 = 0 gives d = 10 or d = -14; both are real, so both common differences are allowed.
A quadratic can hold two answers, and both must be kept because each gives a genuinely different progression.
8.EE.A.2Convert To AlgebraPick the smallest third term
The third term is 29 + 2d: d = 10 gives 49, while d = -14 gives 1, so the smallest possible value is 1.
'Smallest possible' means compare every valid case, and a negative common difference can beat a positive one.
7.EE.B.4Extreme PrincipleWhen a problem asks for the smallest possible value, solve for every case first — a negative common difference can give a smaller answer than the obvious positive one.
- Name the common difference
- Use the geometric-progression rule
- Solve for the common difference
- Pick the smallest third term