1/1乘3+1/2乘4+1/3乘5+.+1/1999乘2001等于多少,最好把规律写出来

Question

1/1乘3+1/2乘4+1/3乘5+.+1/1999乘2001等于多少,最好把规律写出来

Answer 1

原式=1/1×3+1/2×4+1/3×5+...+1/1999×2001
=1/2(1/1-1/3)+1/2(1/2-1/4)+1/2(1/3-1/5)+...+1/2(1/1999-1/2001)
=1/2(1/1-1/3+1/3-1/5+...+1/1999-1/2001++1/2(1/2-1/4+1/4-1/6+...+1/1998-1/2000)
=1/2(1-1/2001)+1/2(1/2-1/2000)
=1/2×2000/2001+1/2×999/2000
=1000/2001+999/4000
=(1000×4000+999×2001)/2001×4000
=(4000000+1998999)/8004000
=5998999/8004000
注：1/k(n+k)=1/n[1/k-1/(n+k)]