HMMT 二月 2004 · 冲刺赛 · 第 28 题

英文解析

28. Find the value of
    ( ) ( ) ( ) ( )
    2003 2003 2003 2003 + + + · · · + .
    1 4 7 2002
    2003
    Solution: (2 − 2) / 3
    √
    Let ω = − 1 / 2+ i 3 / 2 be a complex cube root of unity. Then, by the binomial theorem,
    we have
    ( ) ( ) ( ) ( )
    2003 2003 2003 2003
    2 2003 2 3 4 2005
    ω ( ω + 1) = ω + ω + ω + · · · + ω
    0 1 2 2003
    ( ) ( ) ( ) ( )
    2003 2003 2003 2003
    2003
    2 = + + + · · · +
    0 1 2 2003
    ( ) ( ) ( ) ( )
    2003 2003 2003 2003 − 2 − 1 2003 − 2 − 3 − 4 − 2005
    ω ( ω + 1) = ω + ω + ω + · · · + ω
    0 1 2 2003
    ( )
    2003
    If we add these together, then the terms for n ≡ 1 (mod 3) appear with coefficientn
    3, while the remaining terms appear with coefficient 1 + ω + ω = 0. Thus the desired 2
    2 2003 2003 − 2 − 1 2003 2
    sum is just ( ω ( ω + 1) + 2 + ω ( ω + 1) ) / 3. Simplifying using ω + 1 = − ω − 1 2003 2003
    and ω + 1 = − ω gives ( − 1 + 2 + − 1) / 3 = (2 − 2) / 3.