23.(2010•汕头)阅读下列材料:1×2 = (1×2×3-0×1×2),2×3 = (2×3×4-1×2×3),3×4 = (3×4×5-2×3×4),由以上三个等式相加,可得1×2+2×3+3×4= ×3×4×5 = 20.读完以上材料,请你计算下列各题:(1)
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/14 12:18:19
![23.(2010•汕头)阅读下列材料:1×2 = (1×2×3-0×1×2),2×3 = (2×3×4-1×2×3),3×4 = (3×4×5-2×3×4),由以上三个等式相加,可得1×2+2×3+3×4= ×3×4×5 = 20.读完以上材料,请你计算下列各题:(1)](/uploads/image/z/604839-39-9.jpg?t=23%EF%BC%8E%EF%BC%882010%26%238226%3B%E6%B1%95%E5%A4%B4%EF%BC%89%E9%98%85%E8%AF%BB%E4%B8%8B%E5%88%97%E6%9D%90%E6%96%99%EF%BC%9A1%C3%972+%3D+%281%C3%972%C3%973%EF%BC%8D0%C3%971%C3%972%29%2C2%C3%973+%3D+%282%C3%973%C3%974%EF%BC%8D1%C3%972%C3%973%29%2C3%C3%974+%3D+%283%C3%974%C3%975%EF%BC%8D2%C3%973%C3%974%29%2C%E7%94%B1%E4%BB%A5%E4%B8%8A%E4%B8%89%E4%B8%AA%E7%AD%89%E5%BC%8F%E7%9B%B8%E5%8A%A0%2C%E5%8F%AF%E5%BE%971%C3%972%EF%BC%8B2%C3%973%EF%BC%8B3%C3%974%3D+%C3%973%C3%974%C3%975+%3D+20%EF%BC%8E%E8%AF%BB%E5%AE%8C%E4%BB%A5%E4%B8%8A%E6%9D%90%E6%96%99%2C%E8%AF%B7%E4%BD%A0%E8%AE%A1%E7%AE%97%E4%B8%8B%E5%88%97%E5%90%84%E9%A2%98%EF%BC%9A%281%29)
23.(2010•汕头)阅读下列材料:1×2 = (1×2×3-0×1×2),2×3 = (2×3×4-1×2×3),3×4 = (3×4×5-2×3×4),由以上三个等式相加,可得1×2+2×3+3×4= ×3×4×5 = 20.读完以上材料,请你计算下列各题:(1)
23.(2010•汕头)阅读下列材料:
1×2 = (1×2×3-0×1×2),
2×3 = (2×3×4-1×2×3),
3×4 = (3×4×5-2×3×4),
由以上三个等式相加,可得
1×2+2×3+3×4= ×3×4×5 = 20.
读完以上材料,请你计算下列各题:
(1) 1×2+2×3+3×4+•••+10×11(写出过程);
(2) 1×2+2×3+3×4+•••+n×(n+1) = _________;
(3) 1×2×3+2×3×4+3×4×5+•••+7×8×9 = _________.
第(1)和(2) 不用 直接写第三
23.(2010•汕头)阅读下列材料:1×2 = (1×2×3-0×1×2),2×3 = (2×3×4-1×2×3),3×4 = (3×4×5-2×3×4),由以上三个等式相加,可得1×2+2×3+3×4= ×3×4×5 = 20.读完以上材料,请你计算下列各题:(1)
!你题错了!应该是1×2 = 1/3 (1×2×3-0×1×2)=1×2×3*1/3,前面要乘1/3,不信你自己算算.
(3)1×2×3+2×3×4+3×4×5+?+7×8×9 =1/4[ (1×2×3×4-0*1*2*3)+(2×3×4×5-1×2×3×4)+.(7×8×9×10-6×7×8×9)=1/4(7×8×9×10-0*1*2*3)=1260(裂项)
n(n-1)=1/3(n*(n-1)*(n+1)-(n-2)(n-1)n)所以第一问就是1/3*(n*(n-1)*(n+1)),同理第二问是1/4*(7*8*9*10)=1260
1×2+2×3+3×4+•••+10×11
=1×2×3-0×1×2+2×3×4-1×2×3+3×4×5-2×3×4+............+10×11×12-9×10×11
=10×11×12-0×1×2
=132-0
=132
1×2+2×3+3×4+•••+n×(n+1)
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1×2+2×3+3×4+•••+10×11
=1×2×3-0×1×2+2×3×4-1×2×3+3×4×5-2×3×4+............+10×11×12-9×10×11
=10×11×12-0×1×2
=132-0
=132
1×2+2×3+3×4+•••+n×(n+1)
=1×2×3-0×1×2+2×3×4-1×2×3+3×4×5-2×3×4+............+n×(n+1)(n+2)-(n-1)×n×(n+1)
=n×(n+1)(n+2)-0×1×2
=n×(n+1)(n+2)-0
=n(n+1)(n+2)
1*2*3=1/4(1*2*3*4-0*1*2*3)
2*3=1/4(2*3*4*5-1*2*3*4)
3*4=1/4(3*4*5*6-2*3*4*5)
以上三式相加:1*2*3+2*3*4+3*4*5=1/4*3*4*5*6=90
所以1*2*3+2*3*4+3*4*5+……7*8*9=1/4*7*8*9*10=1260
收起
这道题考察了裂项法,裂项相消。
n×(n+1)×(n+2) = [n×(n+1)×(n+2)×(n+3) - (n-1)×n×(n+1)×(n+2)]/4
1×2×3+2×3×4+3×4×5+…+7×8×9
= [1×2×3×4 - 0×1×2×3 + 2×3×4×5 - 1×2×3×4 + … + 7×8×9×10 - 6×7×8×9]/4
= 7×8×9×10/4
= 1260
不会