AMC 10 · 2019 · #22
Grade 8 algebraPick an answer.
Tool #13 (Algebra): factor the numerator to expose the fixed point x = 4, then substitute y_n = x_n - 4 to move the equilibrium to 0. Tool #9 (Easier Problem): near y = 0 the recursion becomes nearly linear (y_n+1 ≈ 9/10 y_n) — a geometric sequence we can solve in closed form. Tool #5 (Pattern): the ratio 9/10 together with the wide answer-choice intervals lets us estimate m by m ≈ 20 log 2 / log(10/9). Tool #6 (Guess and Check): plug m ≈ 130 into the answer-choice intervals. Tool #3 (Eliminate): the wide intervals are robust to crude estimates.
Factor the numerator
Factoring shows four is a fixed point.
Factor reveals the constant target x = 4 that the sequence drifts toward.
8.EE.C.7Convert To AlgebraRewrite in terms of the gap
Rewrite using the gap from four.
Shift coordinates so the equilibrium is at 0 — the recursion becomes a simple ratio.
Shifting the coordinate so the settled value sits at zero turns the rule into a plain ratio.
▸ Why?
Sliding everything by the same amount leaves every gap exactly as it was.
▸ Why?
After the shift each gap is the previous one times a fixed factor, which is what makes it geometric.
How fast the gap shrinks
Each step multiplies the gap by about nine tenths.
Near the fixed point the recursion looks like a geometric sequence with ratio 9/10.
8.F.B.4Solve An Easier Related ProblemEstimate the number of steps
Logarithms estimate the step count.
Solve the geometric-decay equation by taking logs.
8.EE.A.4Look For A PatternEvaluate numerically
That comes out around one hundred thirty.
Plug standard log values to get a numerical estimate.
8.EE.A.4Guess And CheckPick the interval
Choose the interval containing it.
Estimate sits in (C) with plenty of room — even crude estimate suffices.
6.NS.C.7Eliminate PossibilitiesRead the answer
The answer is eighty-one through two hundred forty-two.
Match estimate to the unique containing interval.
6.NS.C.7Eliminate PossibilitiesThis AMC 12 problem only needs Grade 8 logs and geometric sequences — shift coordinates with y_n = x_n - 4 to expose the ratio 9/10, then (9/10)^m ≈ 1/2²⁰ gives m ≈ 131, which lands in [81, 242].