Question

数列（an）的通项an=n^2(cos^2nπ/3-sin^2nπ/3)，其前项和为Sn,则S30为多少

Answer 1

(cos(nπ/3))^2-(sin(nπ/3))^2=cos(2nπ/3)
n=1,cos(2π/3)=-1/2
n=2,cos(4π/3)=-1/2
n=3,cos(6π/3)=1
以后cos取值三个一组循环。

三个一组分析，3n-2、3n-1、3n为一组
(3n)^2×(1)=(3n)^2×(1/2)+(3n)^2×(1/2)
分别与(3n-2)^2×(-1/2)和(3n-1)^2×(-1/2)相加
(3n)^2×(1/2)+(3n-2)^2×(-1/2)
=(1/2)×((3n)^2-(3n-2)^2)
=(1/2)×(3n+(3n-2))×(3n-(3n-2))
=(1/2)×(6n-2)×(2)
=6n-2
另一个同理
(3n)^2×(1/2)+(3n-1)^2×(-1/2)
=(1/2)×((3n)^2-(3n-1)^2)
=(1/2)×(3n+(3n-1))×(3n-(3n-1))
=(1/2)×(6n-1)×(1)
=3n-0.5

(3n-2)^2×(-1/2)+(3n-1)^2×(-1/2)+(3n)^2×(1)
=(1/2)×[(3n)^2-(3n-2)^2]+(1/2)×[(3n)^2-(3n-1)^2]
=(1/2)(3n+3n-2)(3n-3n+2)+(1/2)(3n+3n-1)(3n-3n+1)
=9n-2.5
也可以直接展开

Bn=9n-2.5
B1=A1+A2+A3 B2=A4+A5+A6 ……
S30共10组
S30=B1+B2+……+B10
用等差数列求和公式
B1=9×1-2.5=6.5
B10=9×10-2.5=87.5
S30=(B1+B10)×10/2=(6.5+87.5)×10/2=470

2009年高考江西数学理科选择题第8题，文科倒数第2题。

Answer 2

解：∵数列{a[n]}的通项a[n]=n^2[(cosnπ/3)^2-(sinnπ/3)^2]，前n项和为S[n]
∴a[n]=n^2Cos(2nπ/3)
∴S[n]=1^2(-1/2)+2^2(-1/2)+3^2+4^2(-1/2)+5^2(-1/2)+6^2+...+n^2Cos(2nπ/3)

当n=3k-2，即：k=(n+2)/3时：
S[3k-2]
=(-1/2)[1^2+2^2+...+(3k-2)^2]+3^3/2[1^2+2^2+...+(k-1)^2]
=-(3k-2)(3k-1)(6k-3)/12+27(k-1)k(2k-1)/12
=[-n(n+1)(2n+1)+(n-1)(n+2)(2n+1)]/12
=(2n+1)(-n^2-n+n^2+n-2)/12
=-(2n+1)/6

当n=3k-1，即：k=(n+1)/3时：
S[3k-1]
=(-1/2)[1^2+2^2+...+(3k-1)^2]+3^3/2[1^2+2^2+...+(k-1)^2]
=-(3k-1)3k(6k-1)/12+27(k-1)k(2k-1)/12
=[-n(n+1)(2n+1)+(n-2)(n+1)(2n-1)]/12
=(n+1)(-2n^2-n+2n^2-5n+2)/12
=(n+1)(-6n+2)/12
=-(n+1)(3n-1)/6

当n=3k，即：k=n/3时：
S[3k]
=(-1/2)[1^2+2^2+...+(3k)^2]+3^3/2(1^2+2^2+...+k^2)
=-3k(3k+1)(6k+1)/12+27k(k+1)(2k+1)/12
=[-n(n+1)(2n+1)+n(n+3)(2n+3)]/12
=n(-2n^2-3n-1+2n^2+9n+9)/12
=n(6n+8)/12
=n(3n+4)/6