2011 AMC 10A Problem 4

Question

Let

X

and

Y

be the following sums of arithmetic sequences:

\begin{aligned} X &= 10+12+14+\cdots+100,\\ Y &= 12+14+16+\cdots+102. \end{aligned}

What is the value of

Y - X?

92

98

100

102

112

Most terms in the two sums cancel

Only the first term of

X

and the last term of

Y

do not match

Note that the terms

12

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.

Problem 4 in other years