1+3+5···+2n-3+2n-1

来源:学生作业帮助网 编辑:作业帮 时间:2024/06/26 02:18:20
1+3+5···+2n-3+2n-1

1+3+5···+2n-3+2n-1
1+3+5···+2n-3+2n-1

1+3+5···+2n-3+2n-1
1+3+5…+2n-3+2n-1
原式 =(2×1-1)+(2×2-1)+(2×3-1).+[2(n-1)-1]+(2n-1)
=2×1-1+2×2-1+2×3-1.+2(n-1)-1+2n-1
=2×1+2×2+2×3.+2(n-1)+2n-1×n
=2×(1+2+3...+n-1+n)-n
=2×[(1+n)×n÷2]-n
=2×(n+n的二次方)÷2-n
=n+n的二次方-n
=n的二次方

(1+3+5···+2n-3+2n-1 +
2n-1+2n-3+...+5+3+1)/2
(2n)*n/2=n^2
或者利用数列的公式:
(1+2n-1)*n/2=n^2

原式 =(2×1-1)+(2×2-1)+(2×3-1)....+[2(n-1)-1]+(2n-1)
=2×1-1+2×2-1+2×3-1....+2(n-1)-1+2n-1
=2×1+2×2+2×3....+2(n-1)+2n-1×n
=2×(1+2+3...+n-1+n)-n
=2×[(1+n)×n÷2]-n
=2×(n+n^2)÷2-n
=n+n^2-n
=n^2