1⼀1*2*3+1⼀2*3*4+1⼀3*4*5+1⼀4*5*6 解题技巧

1/((n-1)*n*(n+1))=0.5*(1/(n-1)*n-1/n*(n+1))=0.5*[(1/(n-1)-1/n)-(1/n-1/(n+1))]=0.5*(1/(n-1)+1/(n+1)-2/n)
1/1*2*3+1/2*3*4+1/3*4*5+1/4*5*6
=1/2(1+1/3-2/2+1/2+1/4-2/3+1/3+1/5-2/4+1/4+1/6-2/5)
=1/2*7/15
=7/30

每项都有1/3 和1/2 提个1/2*1/3出来
那么原式=1/6（1+1/4 +1/10 +1/20）= 1/6*（（20+5+2+1）/20）=1/6 *28/20=28/120=7/30

1/1*2*3+1/2*3*4+1/3*4*5+1/4*5*6
=1/2*(1/1*2-1/2*3)+1/2*(1/2*3-1/3*4)+1/2*(1/3*4-1/4*5)+1/2*(1/4/5-1/5*6)
=1/2*(1/1*2-1/5*6)
=7/30

1/(1×2×3)+1/(2×3×4)+1/(3×4×5)+1/(4×5×6)
=(1/2)[1/(1×2)-1/(2×3)+1/(2×3)-1/(3×4)+1/(3×4)-1/(4×5)+1/(4×5)-1/(5×6)]
=(1/2)[1/(1×2)-1/(5×6)]
=(1/2)(1/2-1/30)
=(1/2)(14/30)
=7/30

1/1*2*3+1/2*3*4+1/3*4*5+1/4*5*6
=(1-1/2-1/3)+(1/2-1/3-1/4)+(1/3-1/4-1/5)+(1/4-1/5-1/6)
=1-1/2-1/3+1/2-1/3-1/4+1/3-1/4-1/5+1/4-1/5-1/6
=1-1/3-1/4-1/6
=12/12-4/12-3/12-2/12
=1/4