AMC 10 · 2002 · #23

Grade 6 arithmetic
Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Figure-N Growth Sequence
recursive-sequencetriangular-numberspattern-recognition pattern-recognition ↑ Prerequisites: recursive-sequence
📏 Long solution 💡 3 insights
Problem
A list of integers starts with a₁ = 1, and aₘ₊ₙ = aₘ + aₙ + mn for every pair of positive integers m and n. Find the twelfth term, a₁₂.

Pick an answer.

(A)
45
(B)
56
(C)
67
(D)
78
(E)
89

AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

The rule a_m+n=a_m+a_n+mn has two free letters, which looks heavy. Tool #9 (Solve an Easier Related Problem) tames it: since the rule holds for every m and n, set the easiest value n=1, turning the two-letter rule into a plain step-by-step recipe for getting the next term. Tool #5 (Look for a Pattern) then does the real work — list the first few terms the recipe produces and recognize them as the familiar triangular numbers 1,3,6,10,…, whose kth value is 1+2+…+k. Tool #4 (Introduce a Variable) keeps the bookkeeping honest by tracking the index m as we climb from a₁ up to a₁₂.

1STEP 1

Set n=1 to get a step rule

The rule holds for every m and n, so take n=1. With a₁ = 1 it becomes aₘ₊₁ = aₘ + m + 1: each next term adds m+1.

a_m+1 = a_m + a₁ + m = a_m + (m+1)
2STEP 2

Build the terms and spot the pattern

Climb from a₁ = 1: a₂ = 3, a₃ = 6, a₄ = 10, a₅ = 15 — the triangular numbers, so aₖ = 1 + 2 + … + k.

a₁=1, a₂=3, a₃=6, a₄=10, a₅=15 → a_k = 1+2+…+k
3STEP 3

Add up to the twelfth term

So a₁₂ = 1 + 2 + … + 12. Pair the ends — 1+12, 2+11, … — six pairs of 13, so a₁₂ = 6 × 13 = 78, choice (D).

a₁₂ = 1+2+…+12 = 6 × 13 = 78 → (D)
Answer
78
Check 78 against the original two-letter rule without the n=1 shortcut. Take m=n=6: a₁₂ = a₆ + a₆ + 6 · 6 = 2a₆ + 36. From the list a₆ = 21, so a₁₂ = 2 · 21 + 36 = 42 + 36 = 78 — the same answer by a completely different split, which is strong confirmation. The value is also sensible in size: a₁₂ should be near 1+2+…+12, comfortably between the small decoy 45 and the large decoy 89.
💡Key takeaway

When a rule works for every input, plug in the easiest one — here n=1 turns the rule into 'add the next number,' building the triangular numbers straight up to 78.

  • Set n=1 to get a step rule
  • Build the terms and spot the pattern
  • Add up to the twelfth term

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