2016 AMC 8 Problems/Problem 8

Find the value of the expression $100-98+96-94+92-90+\cdots+8-6+4-2.$ $\textbf{(A) }20\qquad\textbf{(B) }40\qquad\textbf{(C) }50\qquad\textbf{(D) }80\qquad \textbf{(E) }100$

Solution

We can group each subtracting pair together: $(100-98)+(96-94)+(92-90)+ \ldots +(8-6)+(4-2).$ After subtracting, we have: $2+2+2+\ldots+2+2=2(1+1+1+\ldots+1+1).$ There are 50 even numbers, therefore there are $50/2=25$ even pairs. Therefore the sum is $2 \cdot 25=\boxed{\textbf{(C) }50}$

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2016 AMC 8 Problems/Problem 8

Solution