&1 - 2 + 3 - 4 + … - 98 & + 99 = ?

1999 AMC 12 Problem 1

Attempt Problem 1 of the 1999 AMC 12 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 1999 AMC 12 solutions, or check the answer key.

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\begin{aligned} &1 - 2 + 3 - 4 + \cdots - 98 \\ &\quad {}+ 99 = \, ? \end{aligned}

-50

-49

0

49

50

Group the terms in consecutive pairs

Each pair

(2k-1) - 2k

equals

-1

Pairing consecutive terms gives

\begin{aligned} &(1-2) + (3-4) + \cdots \\ &\quad {}+ (97-98) + 99. \end{aligned}

There are

49

pairs, each equal to

-1,

so the sum is

-49 + 99 = 50.

Thus, the correct answer is E.