AMC 10 · 2009 · #25
Grade 11 algebrapatternPick an answer.
Computing 2009 terms of (a_n+a_n+1)/(1-a_na_n+1) by hand is out of the question, and the terms themselves are ugly numbers in √3. Tool #15 (Organize Information in More Ways) supplies the one observation that changes everything: that fraction is the tangent addition formula, so the rule is really "add the two previous angles". Tool #4 (Introduce a Variable) makes that precise by naming an angle θ_n for each term, which converts a messy multiplicative recursion into plain addition. Tool #9 (Solve an Easier Related Problem) then shrinks the problem twice: angles only matter modulo 180° because tan repeats, and every angle turns out to be a multiple of 15°, so the whole sequence collapses to a Fibonacci-style recursion on the twelve residues mod 12. Tool #2 (Make a Systematic List) writes that short list out and exposes the cycle, and Tool #5 (Look for a Pattern) converts the cycle length into one division with remainder. Tool #3 (Eliminate Possibilities) does the load-bearing safety check that the slick substitution usually hides: 90° is the single angle with no tangent, and the argument is only valid because that value never occurs — which is also the reason the original denominator is never 0.
The rule is the tangent addition formula
The rule is the tangent addition formula.
That fraction is not a random algebraic mess; it is the exact price you pay for adding two angles and then taking the tangent.
9.A-SSE.A.2Organize Information In More WaysBuild the angle list on its own
Tracking angles instead of terms simplifies everything.
Angles add cleanly where tangents do not, and everything in sight is a multiple of 15°, so only twelve angles can ever show up.
9.F-IF.A.3Introduce A VariableThe list closes after 24 terms
The angle list repeats after a fixed number of steps.
Two consecutive entries decide the entire future, so the instant the starting pair shows up again the whole list restarts.
Two consecutive entries decide the whole future, so the moment the starting pair reappears the list restarts.
▸ Why?
Once a state repeats, everything that followed it the first time follows it again in the same order.
▸ Why?
Inside a loop of fixed length only the remainder after dividing by that length decides where a step lands.
Why 6 never appears, and why that matters
One forbidden angle never appears, so the rule never breaks.
90° is the one angle a tangent cannot reach, so the whole method survives only if the angle list steps around it — and it does.
9.F-IF.A.1Eliminate PossibilitiesThe substitution becomes a theorem
That turns the substitution into a genuine theorem.
One fact does two jobs: no angle equals 90°, so the addition formula is legal and the denominator stays alive.
11.F-TF.C.9Introduce A VariableLand on the 2009th term
The remainder lands the target term on 0.
Once the cycle length is known, a two-thousand-step question shrinks to one division with remainder.
6.NS.B.2Look For A PatternCheck it with no trigonometry at all
Direct computation confirms 0, choice (A).
If the angle story is true, the raw numbers have to come back to the start after 24 steps as well — and they do.
11.N-RN.A.2Make A Systematic ListWhen a rule looks like (x+y)/(1-xy), it is angles being added in disguise — trade each number for its angle and a 2009-step monster turns into a list that repeats every 24.
- The rule is the tangent addition formula
- Build the angle list on its own
- The list closes after 24 terms
- Why 6 never appears, and why that matters
- The substitution becomes a theorem
- Land on the 2009th term
- Check it with no trigonometry at all