AMC 10 · 2002 · #21
Grade 6 patternPick an answer.
Adding two thousand terms by hand is out of the question, and the answer choices sit too close together for estimation alone to separate them. Tool #2 (Make a Systematic List) writes out terms until something repeats. Tool #5 (Look for a Pattern) turns that repetition into a proof rather than a hope: since every term is decided by the two before it, the moment a consecutive pair reappears the entire sequence repeats forever. Tool #9 (Solve an Easier Related Problem) then replaces summing thousands of terms with summing one short block and multiplying. Only the last stretch is left, where the running total crosses 10,000 partway through a block, and that is short enough for tool #6 (Guess and Check), stepping one term at a time.
Write terms until a pair repeats
Writing terms by the rule, the opening pair 4,7 returns at positions 13 and 14, so the block is 12 long.
Only the last two terms matter, so once that pair comes back the sequence has forgotten everything else.
4.NBT.B.4Make A Systematic ListProve the block repeats forever
Equal pairs force identical futures, so the sequence is periodic from the very first term.
Two identical starting pairs must produce two identical futures, because the rule looks nowhere else for information.
Once a pair of consecutive terms repeats, the whole sequence must repeat from there on for ever.
▸ Why?
The rule looks only at the last two terms, so identical pairs must be followed by identical futures.
▸ Why?
There are only finitely many possible pairs of digits, so some pair is bound to turn up a second time.
Sum one block, then many blocks
One block sums to 60, so k whole blocks give 60k over 12k terms.
Identical blocks mean the running total climbs by the same 60 every twelve steps, like counting by sixties.
4.NBT.B.4Solve An Easier Related ProblemFind the last full block below target
Since 10,000 = 60 · 166 + 40, the last full block ends at S₁₉₉₂ = 9960, still short.
Whole blocks get you close in one multiplication; the leftover 40 has to be walked off by hand.
6.NS.B.2Solve An Easier Related ProblemWalk the next block term by term
Stepping into the next block, S₁₉₉₈ = 9996 and S₁₉₉₉ = 10,002, so the answer is 1999, choice (B).
Once the gap is only 40, you just add digits one at a time and watch for the step that closes it.
4.NBT.B.4Guess And CheckWhen a rule only looks at the last two terms, a repeated pair means the whole sequence loops forever — so add up one loop, multiply, and then walk the leftover terms by hand.
- Write terms until a pair repeats
- Prove the block repeats forever
- Sum one block, then many blocks
- Find the last full block below target
- Walk the next block term by term