2008 AMC 10B Problem 11

Attempt Problem 11 of the 2008 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 10B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

11.

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?

40

53

68

88

104

Write

u_5

and

u_6

in terms of

u_4

and

u_3

u_5=2u_4+9

and

u_6=2u_5+u_4,

128=5u_4+18

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.

Problem 11 in other years