作业帮两题简便运算
来源：学生作业帮 编辑：拍题作业网作业帮 分类：数学作业 时间：2024/05/01 07:17:42
两题简便运算
1=2x(1-1/2)
1/(1+2)=2x(1/2-1/3)
1/(1+2+3)=2x(1/3-1/4)
1/(1+2+3+4)=2x(1/4-1/5)
……
1/(1+2+3+……+100)=2x(1/100-1/101)
所以：
1+1/(1+2)+1/(1+2+3)+……+1/(1+2+3+……+100)
=2x(1-1/2+1/2-1/3+1/3-1/4+……-1/100+1/100-1/101)
=2x(1-1/101)
=200/101 再答： （1-1/2^2）*（1-1/3^3）*（1-1/4^4）*···*（1-1/10^10） =(1+1/2)*(1-1/2)*(1+1/3)(1-1/3)*(1+1/4)(1-1/4)*...*(1+1/10)*(1-1/10) =(3/2)*(1/2)*(4/3)*(2/3)*(5/4)*(3/4)*...*(11/10)*(9/10) =[(1/2)*(2/3)*(3/4)*...*(9/10)]*[(3/2)*(4/3)*(5/4)*...*(11/10)] =(1/10)*(11/2) =11/20
再答： （1-1/2^2）*（1-1/3^2）*（1-1/4^2）*···*（1-1/10^2） =(1+1/2)*(1-1/2)*(1+1/3)(1-1/3)*(1+1/4)(1-1/4)*...*(1+1/10)*(1-1/10) =(3/2)*(1/2)*(4/3)*(2/3)*(5/4)*(3/4)*...*(11/10)*(9/10) =[(1/2)*(2/3)*(3/4)*...*(9/10)]*[(3/2)*(4/3)*(5/4)*...*(11/10)] =(1/10)*(11/2) =11/20
1/(1+2)=2x(1/2-1/3)
1/(1+2+3)=2x(1/3-1/4)
1/(1+2+3+4)=2x(1/4-1/5)
……
1/(1+2+3+……+100)=2x(1/100-1/101)
所以：
1+1/(1+2)+1/(1+2+3)+……+1/(1+2+3+……+100)
=2x(1-1/2+1/2-1/3+1/3-1/4+……-1/100+1/100-1/101)
=2x(1-1/101)
=200/101 再答： （1-1/2^2）*（1-1/3^3）*（1-1/4^4）*···*（1-1/10^10） =(1+1/2)*(1-1/2)*(1+1/3)(1-1/3)*(1+1/4)(1-1/4)*...*(1+1/10)*(1-1/10) =(3/2)*(1/2)*(4/3)*(2/3)*(5/4)*(3/4)*...*(11/10)*(9/10) =[(1/2)*(2/3)*(3/4)*...*(9/10)]*[(3/2)*(4/3)*(5/4)*...*(11/10)] =(1/10)*(11/2) =11/20
再答： （1-1/2^2）*（1-1/3^2）*（1-1/4^2）*···*（1-1/10^2） =(1+1/2)*(1-1/2)*(1+1/3)(1-1/3)*(1+1/4)(1-1/4)*...*(1+1/10)*(1-1/10) =(3/2)*(1/2)*(4/3)*(2/3)*(5/4)*(3/4)*...*(11/10)*(9/10) =[(1/2)*(2/3)*(3/4)*...*(9/10)]*[(3/2)*(4/3)*(5/4)*...*(11/10)] =(1/10)*(11/2) =11/20