AMC 10 · 2008 · #11

Grade 8 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

recursive-sequencesystems-of-equationslinear-equations-two-var convert-to-algebra ↑ Prerequisites: systems-of-equations

📏 Medium solution 💡 2 insights

Problem

A sequence of real numbers follows the rule uₙ₊₂=2uₙ₊₁+uₙ: each term is twice the term just before it plus the term before that. Given u₃=9 and u₆=128, find u₅.

Pick an answer.

(A)

$\ 40$

(B)

$\ 53$

(C)

$\ 68$

(D)

$\ 88$

(E)

$\ 104$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The gap between the known term u₃ and the known term u₆ is filled by two hidden terms, u₄ and u₅. Tool #4 (Introduce a Variable) names those two unknowns so we can work with them. Tool #13 (Convert to Algebra) then turns the recurrence into equations: applying the rule once gives a relation among u₃,u₄,u₅, and applying it again gives a relation among u₄,u₅,u₆ — two equations in the two unknowns. Solving that pair pins down u₅. Tool #3 (Eliminate Possibilities) is the safety net: the answer must be one of the five listed values, and the algebra lands on exactly one of them.

1STEP 1

Name the hidden terms

Between u₃=9 and u₆=128 sit two unknown terms: let a=u₄ and b=u₅. The value we want is b.

a=u₄, b=u₅

2STEP 2

First equation from the rule

Put n=3 in the rule: the newest term is u₅, so u₅=2u₄+u₃, which in the new names reads b=2a+9.

u₅=2u₄+u₃ → b=2a+9

3STEP 3

Second equation from the rule

Now n=4 makes u₆ the newest term: u₆=2u₅+u₄, and u₆=128 turns that into 128=2b+a.

u₆=2u₅+u₄ → 128=2b+a

4STEP 4

Solve the two equations

Put a=(b-9)/2 into 128=2b+a and double both sides: 256=5b-9, so b=53, choice (B).

256=5b-9 → 5b=265 → b=53

Answer

Check the whole chain. With u₅=53, the first equation gives u₄=(53-9)/2=22. Then the rule rebuilds the sequence: u₅=2·22+9=53 and u₆=2·53+22=128, which matches the given u₆ exactly. The terms 9,22,53,128 also grow in the roughly-tripling way the rule forces, so 53 sitting between 22 and 128 is the right size. Among the choices 40,53,68,88,104, only 53 makes the rule reproduce u₆=128.

💡Key takeaway

When terms are missing between two you know, name them with letters, write the rule twice to get two equations, and solve.

Name the hidden terms
First equation from the rule
Second equation from the rule
Solve the two equations

Name the hidden terms

First equation from the rule

Second equation from the rule

Solve the two equations

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