AMC 8 Daily Practice - Multiplication Formula

Complete problem set with solutions and individual problem pages

Problem 5 Easy

Calculate the shaded area formed by overlapping squares with side lengths 1, 2, 3, \ldots, 2008.

  • A.

    2008000

  • B.

    2009000

  • C.

    2017036

  • D.

    1000000

  • E.

    2007000

Answer:C

The shaded area is the alternating sum of squares:

(2^2-1^2) + (4^2-3^2) + \cdots + (2008^2-2007^2) \\

= (2+1)(2-1) + (4+3)(4-3) + \cdots + (2008+2007)(2008-2007) \\

= 1 + 2 + 3 + \cdots + 2008 \\

= \frac{2008 \times (1+2008)}{2} = 2017036

Final result: \boxed{2017036}

Related Problems
Problem 3 AMC 8 Daily Practice - Multiplication Formula
Problem 4 AMC 8 Daily Practice - Multiplication Formula
Problem 6 AMC 8 Daily Practice - Multiplication Formula
Problem 7 AMC 8 Daily Practice - Multiplication Formula
Think Academy Powered by ChatGPT