Question

数列{an}的通项an=n(cosnΠ/3^2-sinnΠ/3),其前n项和为Sn,则S30=

Answer 1

Bn=cos²nΠ/3 - sinnΠ/3 具有周期性,周期为6

n=1,B1=1/4-√3/2

n=2,B2=1/4-√3/2

n=3,B3=1

n=4,B4=1/4+√3/2

n=5,B5=1/4+√3/2

n=6,B6=1

共有5个周期,原数列可看作6个等差数列.

S30=5(1+25)*(1/4-√3/2)/2+5(2+26)*(1/4-√3/2)/2

+5(3+27)/2+5(4+28)*(1/4+√3/2)/2+5(5+29)*(1/4+√3/2)/2 +5(6+30)/2

=240+15√3