51问答网 > 计算:3⼀(1*2*3*4)+3⼀(2*3*4*5)+3⼀(3*4*5*6)+...+3⼀(7*8*9*10)

计算:3⼀(1234)+3⼀(2345)+3⼀(3456)+...+3⼀(78910)

用简便算，不要死算！

2025-02-23 17:56:56

推荐回答（1个）

回答1：

an=3/n(n+1)(n+2)(n+3)
=3{[1/n-1/(n+1)][1/(n+2)-1/(n+3)]}
=3{1/n(n+2)-1/n(n+3)-1/(n+1)(n+2)+1/(n+1)(n+3)}
=3{1/2[1/n-1/(n+2)-1/3[1/n-1/(n+3)]-[1/(n+1)-1/(n+2)]+1/2[1/(n+1)-1/(n+3)]}
=3{1/6n-1/2(n+1)+1/2(n+2)-1/6(n+3)}
=1/2[1/n-1/(n+3)]+3/2[1/(n+2)-1/(n+1)]

3/(1*2*3*4)+3/(2*3*4*5)+3/(3*4*5*6)+...+3/(7*8*9*10)
=1/2(1/1+1/2+...+1/7-1/4-1/5-...-1/10)+3/2(1/3+1/4+...+1/9-1/2-1/3-...-1/8)
=1/2(1/1+1/2+1/3-1/8-1/9-1/10)+3/2(-1/2+1/9)
=1/2+1/2(1/2-3/2)+1/6-1/16+1/9(-1/2+3/2)-1/20
=1/6-1/16-1/9-1/20

剩下的慢慢算吧