巧算:1×2+2×3+3×4+...+49×50怎么算?
来源:学生作业帮 编辑:搜狗做题网作业帮 分类:数学作业 时间:2024/06/17 22:11:43
巧算:1×2+2×3+3×4+...+49×50怎么算?
1×2+2×3+3×4+...+49×50怎么算?要求有计算过程.
1×2+2×3+3×4+...+49×50怎么算?要求有计算过程.
![巧算:1×2+2×3+3×4+...+49×50怎么算?](/uploads/image/z/7819043-59-3.jpg?t=%E5%B7%A7%E7%AE%97%EF%BC%9A1%C3%972%2B2%C3%973%2B3%C3%974%2B%EF%BC%8E%EF%BC%8E%EF%BC%8E%2B49%C3%9750%E6%80%8E%E4%B9%88%E7%AE%97%3F)
答案:41650 一般的,(用*表示相乘) 设S=1*2+2*3+3*4+……n*(n+1) 则3S=1*2*3+2*3*3+3*4*3+4*5*3+……n*(n+1)*3 =1*2(3-0)+2*3(4-1)+3*4(5-2)+……n(n+1)[n+2-(n-1)] =1*2*3+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), 所以原式=S=n(n+1)(n+2)/3 故S=49*(49+1)*(49+2)/3=41650