AMC 10 · 2011 · #17

Grade 6 patternalgebra
Archetype Pair-Sum Invariant └ Sub-archetype Universe Total Minus Known
recursive-sequencesystems-of-equations pattern-recognitionconvert-to-algebra ↑ Prerequisites: pattern-recognition
📏 Medium solution 💡 2 insights
Problem
Eight terms sit in a row: A, B, C, D, E, F, G, H. The third term is C=5, and every block of three terms that are next to each other adds up to 30. Find the sum of the first and last terms, A+H.

Pick an answer.

(A)
17
(B)
18
(C)
25
(D)
26
(E)
43

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

The 'every three neighbors sum to 30' rule links overlapping groups. If I write two overlapping groups as equations and compare them, a repeating pattern pops out: the sequence cycles every three terms. Once I see that cycle I can jump straight from C and the sum rule to A+H without ever finding the individual terms.

1STEP 1

Write the sum rule with letters

Name the terms, then write the 30-sum rule for the first group and again after sliding over one term.

A+B+C = 30 B+C+D = 30
2STEP 2

Compare the overlapping groups

Both groups equal 30 and share B+C, so the leftovers match: A=D. Sliding on gives B=E, then C=F.

A + (B+C) = D + (B+C) → A = D
3STEP 3

Spot the repeat-every-three pattern

Each term equals the one three places later, so the row cycles A, B, C — and the eighth slot H lands on B.

A, B, C, A, B, C, A, B → H = B
4STEP 4

Turn A+H into A+B and finish

Since H=B, A+H is just A+B, and A+B+C=30 with C=5 gives A+H = 25.

A+H = A+B = 30 - C = 30 - 5 = 25
Answer
25
Test the pattern with a concrete sequence. Pick A=10, B=15; then C=5 makes A+B+C=30, and the cycle gives 10,15,5,10,15,5,10,15. Every three neighbors do sum to 30, and A+H = 10+15 = 25, matching (C). Choosing different starting values (say A=0, B=25) still gives A+H=25, confirming the answer does not depend on which valid terms are used.
💡Key takeaway

When overlapping groups all add to the same total, the sequence repeats, so a far-away term is really a near one in disguise.

  • Write the sum rule with letters
  • Compare the overlapping groups
  • Spot the repeat-every-three pattern
  • Turn A+H into A+B and finish

More like this

Same archetype — closest grade level first.