找规律

题目

找规律
1/2=1/1×1/2=1/1-1/2;1/6=1/(2×3)=1/2-1/3;1/12=1(3×4)=1/3-1/4
(1)用含n（n为正整数）的等式表示
（2）用得到的结论计算
[1/(a-1)(a-2)]+[1/(a-2)(a-3)]+[1/(a-3)(a-4)]+[1/(a-4)(a-5)]

答案

1/2=1/1×1/2=1/1-1/2; 1/6=1/(2×3)=1/2-1/3; 1/12=1(3×4)=1/3-1/4 (1）用含n（n为正整数）的等式表示:1/[n * (n + 1)] = 1/n - 1/(n+1) 或者 1/[(n - 1) * n] = 1/(n - 1) - 1/n (2)用得到的结论计算:[1/(a-1)(a-2)]+[1/(a-2)(a-3)]+[1/(a-3)(a-4)]+[1/(a-4)(a-5)] = [1/(a-2) - 1/(a-1)] + [1/(a-3) - 1/(a-2)] + [1/(a-4) - 1/(a-3)] + [1/(a-5) - 1/(a-4)] = 1/(a-2) - 1/(a-1) + 1/(a-3) - 1/(a-2) + 1/(a-4) - 1/(a-3) + 1/(a-5) - 1/(a-4) = 1/(a-5) - 1/(a-1) = 4/[(a-1)(a-5)]