AMC 10 · 2011 · #17

Grade 6 arithmetic

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

In the eight term sequence $A$ , $B$ , $C$ , $D$ , $E$ , $F$ , $G$ , $H$ , the value of $C$ is $5$ and the sum of any three consecutive terms is $30$ . What is $A+H$ ?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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, $A+H$.

Givens: The sequence has eight terms $A, B, C, D, E, F, G, H$; $C = 5$; Any three consecutive terms sum to $30$

Unknowns: The value of $A + H$

Understand

Givens: The sequence has eight terms $A, B, C, D, E, F, G, H$; $C = 5$; Any three consecutive terms sum to $30$

Plan

Primary tool: #5 Look for a Pattern

Secondary: #4 Introduce a Variable, #15 Organize Information in More Ways

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.

Execute — Answer: C

#4 Introduce a Variable 6.EE.B.6 Step 1

Write the sum rule with letters

Give each term its letter and write down what 'three neighbors add to 30' means for the first two overlapping groups.
The first three terms give one equation; sliding over by one term gives the next.

$$A+B+C = 30 \qquad B+C+D = 30$$

💡 Naming the terms with letters turns the word rule into equations I can line up and compare.

#4 Introduce a Variable 6.EE.A.4 Step 2

Compare the overlapping groups

Both groups equal $30$, and both contain the shared piece $B+C$.
Since $A+B+C$ and $B+C+D$ name the same number and share $B+C$, the leftover parts must match: $A=D$.
The same comparison on the next pair of groups gives $B=E$, then $C=F$, and so on.

$$A + (B+C) = D + (B+C) \;\Rightarrow\; A = D$$

💡 If two equal totals share a common part, the parts left over have to be equal too.

#5 Look for a Pattern 4.OA.C.5 Step 3

Spot the repeat-every-three pattern

Because each term equals the term three places later, the sequence just cycles through $A, B, C$ over and over.
Writing all eight terms out by position makes the pattern plain, and it shows where $H$ lands.

$$A,\ B,\ C,\ A,\ B,\ C,\ A,\ B \;\Rightarrow\; H = B$$

💡 A rule that ties every term to the one three steps back forces the list to loop in threes.

#4 Introduce a Variable 6.EE.B.7 Step 4

Turn A+H into A+B and finish

Since $H=B$, the target $A+H$ is the same as $A+B$.
But $A+B+C=30$ and $C=5$, so $A+B = 30-5 = 25$.
Therefore $A+H = 25$, which is choice (C).

$$A+H = A+B = 30 - C = 30 - 5 = 25$$

💡 Once $H$ is really just $B$ in disguise, the sum rule on the very first group hands me $A+B$ directly.

[1] #4 6.EE.B.6 Give each term its letter and write down what 'three neighbors add to 30' means

[2] #4 6.EE.A.4 Both groups equal $30$, and both contain the shared piece $B+C$. Since $A+B+C$ a

[3] #5 4.OA.C.5 Because each term equals the term three places later, the sequence just cycles t

[4] #4 6.EE.B.7 Since $H=B$, the target $A+H$ is the same as $A+B$. But $A+B+C=30$ and $C=5$, so

Review

Reasonableness: 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.

Alternative: Skip the pattern and use only the first group: $A+B+C=30$ with $C=5$ gives $A+B=25$. Then note the sequence repeats every three terms, so term $8$ (which is $2$ past a multiple of $3$) equals term $2 = B$, hence $H=B$ and $A+H=A+B=25$. Same result by focusing on the single equation that already contains what is asked.

CCSS standards used (min grade 6)

6.EE.B.6 Use variables to represent numbers and write expressions to solve problems (Naming the eight terms with letters and writing the 'sum is 30' rule as equations.)
6.EE.A.4 Identify when two expressions are equivalent (Comparing two overlapping groups that both equal 30 to conclude $A=D$.)
4.OA.C.5 Generate a number or shape pattern following a given rule (Extending the repeat-every-three cycle to place term $H$ and see $H=B$.)
6.EE.B.7 Solve real-world problems by writing and solving equations of the form px = q (Solving $A+B+C=30$ with $C=5$ to get $A+B=25$, which equals $A+H$.)

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

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Write the sum rule with letters

Compare the overlapping groups

Spot the repeat-every-three pattern

Turn A+H into A+B and finish

Review

CCSS standards used (min grade 6)

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