作业帮1×7+2×10+3×13+……+1000×3004
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/04 14:47:02
1×7+2×10+3×13+……+1000×3004
1×7+2×10+3×13+……+1000×3004
1×7+2×10+3×13+……+1000×3004
=1×(1×3+4)+2×(2×3+4)+3×(3×3+4)+……+1000×（1000×3+4)
=3×(1²+2²+3²+……+1000²)+4(1+2+3+……+1000)
=3×1000×(1000+1)(2000+1)/6+4×(1+1000)×1000/2
=1001500500+2002000
=1003502500
两个公式1²+2²+……+n²=n(n+1)(2n+1)/6
1+2+……+n=n(n+1)/2
1×7+2×10+3×13+……+1000×3004
=1×(3+4)+2×(6+4)+3×(9+4)+……+1000×(3000+4)
=1x3+2x6+3x9+...+1000x3000+4x(1+2+3+...+1000)
=3x(1²+2²+3²+...+1000²)+4X500500
=3X1000X1001X200...
全部展开
1×7+2×10+3×13+……+1000×3004
=1×(3+4)+2×(6+4)+3×(9+4)+……+1000×(3000+4)
=1x3+2x6+3x9+...+1000x3000+4x(1+2+3+...+1000)
=3x(1²+2²+3²+...+1000²)+4X500500
=3X1000X1001X2001/6+2002000
=1001500500+2002000
=1003502500
用到以下公式
1+2+3+...+n=n(n+1)/2
1^2+2^2+3^2+…+n^2=n(n+1)(2n+1)/6
收起
1×7+2×10+3×13+……+1000×3004的通项为an=nx(3n+4)=3n²+4n
所以1×7+2×10+3×13+……+1000×3004
=S1000
=3（1²+2²+3²+....+1000²)+4(1+2+3+.....+1000)
=3x1/6x1000x(1000+1)(2X1000+1)+4X(1+1000)X1000/2
=1001500500+2002000
=1003502500
s=1×7+2×10+3×13+……+nx(3n+4)
=3x1^2+3x2^2+3x3^2+……3xn^2+4x1+4x2+4x3+……4xn
=3n(n+1)(2n+1)/6+4n(n+1)/2
=(2n^2+5n)(n+1)/2
把n=1000代入上式的s=1003502500