2008 AMC 10B Problem 11

P11-15~2 minPrint

Sequences & Series Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

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 $u_5$ ?

Show solution without answering

Key insight

The missing term between the two known ones is the fourth term. Call it a, build the fifth and sixth terms from it, and set the sixth equal to 128.

Solution

The rule says each term is twice the term before it plus the term before that. To use it we need two terms in a row, but we know only the 3rd and the 6th. So give the 4th term a name and build forward.

1. Let $u_4 = a$ . We know $u_3 = 9$ .
2. The rule with $n = 3$ : $u_5 = 2u_4 + u_3 = 2a + 9$ .
3. The rule with $n = 4$ : $u_6 = 2u_5 + u_4 = 2(2a + 9) + a = 5a + 18$ .
4. We are told $u_6 = 128$ , so
$5a + 18 = 128, \qquad 5a = 110, \qquad a = 22.$
5. Then $u_5 = 2(22) + 9 = 53$ .
6. Check: $u_6 = 2(53) + 22 = 106 + 22 = 128$ . Correct.

The answer is $\boxed{\textbf{(B)}\ 53}$ .

Why this works

A rule that uses the two previous terms needs two terms in a row to get going. When the known terms are spread apart, name one missing term with a letter and write the later terms using it. The rule then turns into an ordinary one-letter equation.

Alternative approach

Test the answer choices. If $u_5 = x$ , then from $u_6 = 2u_5 + u_4$ we get $u_4 = 128 - 2x$ , and the rule also needs $u_5 = 2u_4 + 9$ . Try (B) $x = 53$ : $u_4 = 128 - 106 = 22$ , and $2(22) + 9 = 53$ . It fits. (Trying (A) $x = 40$ : $u_4 = 48$ , but $2(48) + 9 = 105$ , not $40$ .)

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

The unknown term between the two given ones is $u_4$ ; call it $a$ . Apply the recurrence twice:
$u_5 = 2u_4 + u_3 = 2a + 9,$
$u_6 = 2u_5 + u_4 = 2(2a + 9) + a = 5a + 18.$
Setting $u_6 = 128$ gives $5a + 18 = 128$ , so $a = 22$ . Then
$u_5 = 2(22) + 9 = 53.$
Check: $u_6 = 2(53) + 22 = 128$ .

The answer is $\boxed{\textbf{(B)}\ 53}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 21: Sequences & Series (§21.1, §21.2, §21.3, §21.4)
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Intermediate Algebra, Chapter 17: More Sequences and Series (§17.1, §17.2)
Introduction to Algebra, Chapter 7: Proportion (§7.4)