2011 AMC 10A Problem 4

Below is the professionally curated solution for Problem 4 of the 2011 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 10A solutions, or check the answer key.

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Concepts:arithmetic sequence summation

Difficulty rating: 770

Let X and Y be the following sums of arithmetic sequences: $\begin{align*}X &= 10+12+14+\cdots+100,\\ Y &= 12+14+16+\cdots+102.\end{align*}$ What is the value of $Y - X?$

$92$

$98$

$100$

$102$

$112$

Solution:

Note that the terms $10$ through $100$ are common to both sums. When we subtract, all these terms cancel out.

This means that $Y - X = 102 - 10 = 92.$

Thus, A is the correct answer.

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