2018 AMC 8 Problem 5

Below is the video solution and professionally curated solution for Problem 5 of the 2018 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 8 solutions, or check the answer key.

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Concepts:arithmetic sequence pairing and grouping

Difficulty rating: 870

What is the value of $\begin{align*} &1+3+5+\cdots+2017+2019 \\ -&2-4-6-\cdots-2016-2018? \end{align*}$

$-1010$

$-1009$

$1008$

$1009$

$1010$

Video solution:

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Written solution:

Rearranging the terms, notice that the expression in the question is equal to: $\begin{align*}&1 + (3-2) + (5-4) + \cdots +\\ &(2017-2016) + (2019-2018). \end{align*}$ Each term is equal to $1,$ and there are $\frac{2019-1}2+1 = 1010$ terms, so the total sum is $1010\cdot1 = 1010.$

Thus, E is the correct answer.

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