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1999 AHSME Problems/Problem 1

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Problem

1 - 2 + 3 -4 + \cdots - 98 + 99 =

\mathrm{(A) \ -50 } \qquad \mathrm{(B) \ -49 } \qquad \mathrm{(C) \ 0 } \qquad \mathrm{(D) \ 49 } \qquad \mathrm{(E) \ 50 }

Solution

Solution 1

If we group consecutive terms together, we get (-1) + (-1) + \cdots + 99, and since there are 49 pairs of terms the answer is -49 + 99 = 50 \Rightarrow \mathrm{(E)}.

Solution 2

Let 1 - 2 + 3 -4 + \cdots - 98 + 99 = S.

Therefore, S=99-98+97-\cdots -4+3-2+1

We add:

2S=100-100+100-100\cdots +100=100

S=50\Rightarrow \mathrm{(E)}

See also

1999 AHSME (Problems)
Preceded by
First question
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Want to learn how to tackle those tough AMC/AIME/Olympiad counting and probability problems? Check out Art of Problem Solving's Intermediate Counting & Probability by David Patrick.
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