(-1)^n-1 (2n-1)(2n+1)

来源：学生作业帮助网编辑：作业帮时间：2024/04/29 19:58:17

(-1)^n-1 (2n-1)(2n+1)
(-1)^n-1 (2n-1)(2n+1)

(-1)^n-1 (2n-1)(2n+1)
a1 = (-1)^0 * (2-1)(2+1) = 2^2-1 = 3
a2 = (-1)^1 * (3-1)(3+1) = 1-3^2 = -8
a3 = (-1)^2 * (4-1)(4+1) = 4^2-1=15
a4 = (-1)^3 * (5-1)(5+1) = 1-5^2=-24
.
a(n-1) = (-1)^(n-2) * {n^2 - 1}
an = (-1)^(n-1) * {(n+1)^2 - 1}
当n为奇数时：
a1+a2+a3+a4+.+a(n-2)+a(n-1)+an
= {-1+1-1+1+.-1+1-1} + {2^2+4^2+.+(n+1)^2} - {3^2+5^2+.+n^2}
= {-1+1-1+1+.-1+1} + 4{1^2+2^2+.+[(n+1)/2]^2} - {1^2+3^2+5^2+.+n^2}
= 0 + 4*1/6*[(n+1)/2]*[(n+1)/2+1]*[(n+1)+1] - 1/6*n*(n+1)*(2n+1)
= 1/6*[(n+1)]*[(n+1)+2]*[(n+1)+1] - 1/6*n*(n+1)*(2n+1)
= 1/6*(n+1)*(n+3)(n+2) - 1/6*n*(n+1)*(2n+1)
= 1/6*(n+1)*{(n+3)(n+2)-n(2n+1)}
= 1/6*(n+1)*{n^2+5n+6-2n^2-n}
= 1/6(n+1)(6+4n-n^2)
= 1/6(n+1){(n-2)^2+2}
当n为偶数时：
a1+a2+a3+a4+.+a(n-2)+a(n-1)+an
= {-1+1-1+1+.-1+1} + {2^2+4^2+.+n^2} - {3^2+5^2+.+(n+1)^2}
= 0 + 4{1^2+2^2+.+(n/2)^2} + 1 - {1^2+3^2+5^2+.+(n+1)^2}
= 4*1/6*(n/2)*(n/2+1)*(n+1) +1 - 1/6*(n+1)*(n+2)*{2(n+1)+1}
= 1+ 1/6*n*(n+2)*(n+1) - 1/6*(n+1)*(n+2)*(2n+3)
= 1+ 1/6*(n+1)*(n+2) *(n-2n-3)
= 1 - 1/6(n+1)(n+2)(n+3)

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