AMC 10 · 2008 · #25
Grade 11 algebraPick an answer.
The finish is given and the start is wanted, so tool #11 (Work Backwards) sets the direction. Stepping back 99 times by hand is hopeless, so tool #15 (Organize Information in More Ways) does the real work first: stop carrying a_n and b_n as two separate numbers and pack the pair into the single complex number a_n+b_ni. The two-part rule then collapses into one multiplication, and 99 steps become one power. Tool #9 (Solve an Easier Related Problem) shrinks that power: instead of attacking (√(3)+i)⁹⁹, cube the multiplier and watch it turn into 8i. Tool #5 (Look for a Pattern) names the three-step cycle that results, but the cycle here is proved by that one cubing rather than read off a list of sample terms - which matters, because a pattern spotted in the first few terms carries no promise about the hundredth.
Pack the pair into one number
Packing the pair turns the rule into a multiplication.
A rule that mixes two coordinates is usually one multiplication in disguise, once the coordinates ride inside a single number.
11.N-CN.A.1Organize Information In More WaysOne fixed multiplier, one formula
A fixed multiplier gives a closed form with one power.
Multiplying by the same number at every step means the n-th term is that number raised to a power.
Multiplying by the same number at every step means the nth term is that number raised to a power.
▸ Why?
Every term is the previous one multiplied by one fixed factor, which is what makes the run geometric.
▸ Why?
An exponent counts how many times that factor has been used, so n steps is that factor to the nth power.
Cube the multiplier
Cubing the multiplier gives something very simple.
When a huge exponent looks impossible, try small ones first - an ugly number often turns clean after two or three multiplications.
11.N-CN.A.2Solve An Easier Related ProblemTurn 99 into 33 cubes
The large power then collapses into a single term.
99 is a multiple of 3, so the clean cube can be reused all the way up.
8.EE.A.1Solve An Easier Related ProblemCheck the start is forced
The multiplier is nonzero, so the start is forced.
Multiplying by a nonzero number can always be reversed, so the last point remembers exactly one first point.
11.F-BF.B.4Work BackwardsDivide and add the coordinates
Dividing and adding gives 1/2⁹⁸, choice (D).
1/2⁹⁷ is two copies of 1/2⁹⁸, so taking one copy away leaves one.
11.N-CN.A.3Work BackwardsWhen one rule keeps stirring two numbers together, pack the pair into a single number - then a hundred steps become one multiplication you can undo.
- Pack the pair into one number
- One fixed multiplier, one formula
- Cube the multiplier
- Turn 99 into 33 cubes
- Check the start is forced
- Divide and add the coordinates