2009 AMC 12A Problems/Problem 7

Problem

The first three terms of an arithmetic sequence are $2x - 3$ , $5x - 11$ , and $3x + 1$ respectively. The $n$ th term of the sequence is $2009$ . What is $n$ ?

$\textbf{(A)}\ 255 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 1506 \qquad \textbf{(E)}\ 8037$

Solution

As this is an arithmetic sequence, the difference must be constant: $(5x-11) - (2x-3) = (3x+1) - (5x-11)$ . This solves to $x=4$ . The first three terms then are $5$ , $9$ , and $13$ . In general, the $n$ -th term is $1+4n$ . Solving $1+4n=2009$ we get $n=\boxed{502}$ .

2009 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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2009 AMC 12A Problems/Problem 7

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