设函数f(x)=sin(πx/2+π/4)x

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设函数f(x)=sin(πx/2+π/4)x
设函数f(x)=sin(πx/2+π/4)x

设函数f(x)=sin(πx/2+π/4)x
f(2006)+f(2007)+f(2008)+f(2009)
=sin(2006π/2+π/4)+sin(2007x/2+π/4)+f(2004)+f(2005)
=sin(2006π/2+π/4)+sin(2007π/2+π/4)+sin(2004π/2+π/4)+sin(2005π/2+π/4)
=sin(1004π-π+π/4)+sin(1004π-π/2+π/4)+sin(1002π+π/4)+sin(1002+π/2+π/4)
=sin(-3π/4)+sin(-π/4)+sin(π/4)+sin(3π/4)
=0

f(2008)=f(2004)=sin(π2004/2+π/4)=sin(π2006/2+π/4-π)=-f(2006)
类似的f（2009）=-f（2007）所以
f(2006)+f(2007)+f(2008)+f(2009)=0

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