2008 AMC 10B Problem 11

Problem 11 of 25IntermediateAlgebra

Suppose that

(u_n)

is a sequence of real numbers satisfying

u_{n+2}=2u_{n+1}+u_n,

and that

u_3=9

and

u_6=128.

What is

u_5?

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Solution

Using the recurrence,

u_5=2u_4+u_3=2u_4+9

and

u_6=2u_5+u_4

=2(2u_4+9)+u_4

=5u_4+18.

Setting

5u_4+18=128

gives

u_4=22,

u_5=2\cdot 22+9=53.

Thus, the correct answer is B.

Concepts: recursion · system of equations

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.