2010 AMC 12A Problem 10 — Solution | LIVE by Po-Shen Loh

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2010 AMC 12A Problem 10

Attempt Problem 10 of the 2010 AMC 12A 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 2010 AMC 12A solutions, or check the answer key.

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

10.

The first four terms of an arithmetic sequence are $p,$ $9,$ $3p-q,$ and $3p+q.$ What is the $2010$ th term of this sequence?

$8041$

$8043$

$8045$

$8047$

$8049$

Answer: A

Concepts:arithmetic sequence system of equations

Difficulty rating: 1410

Solution:

Consecutive terms differ by a common difference $d.$ From the last two terms, $d=(3p+q)-(3p-q)=2q.$

From the first two terms, $9-p=d=2q,$ and from the second and third, $(3p-q)-9=d=2q.$ Solving this system gives $p=5,$ $q=2,$ and $d=4.$

The $2010$ th term is $\begin{aligned} p+2009d &= 5+2009\cdot4 \\ &= 8041. \end{aligned}$

Thus, A is the correct answer.

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