如果i^2＝－1,则i^3＝i^2·i＝－i,i^4＝i^2·i^2＝(－1)×(－1)＝1
∴i＋i^2＋i^3＋i^4＝i－1－i＋1＝0
而i^5＋i^6＋i^7＋i^8＝i^4·(i＋i^2＋i^3＋i^4)＝1×0＝0
i^9＋i^10＋i^11＋i^12＝i^8·(i＋i^2＋i^3＋i^4)＝1×0＝0
…
i^2009＋i^2010＋i^2011＋i^2012＝i^2008·(i＋i^2＋i^3＋i^4)＝1×0＝0
故原式＝0