The value of the sum `Sigma_(i=1)^(20) i(1/i+1/(i+1)+1/(i+2)+.....+1/(20))` is ______________.

Answer

115

Explanation

Given sum is
`1(1/1+1/2+1/3+…+1/20)+2(1/2+1/3+..+1/20)+3(1/3+..+1/20)+20(1/20)`
`=1/1(1)+1/2(1+2)+1/3(1+2+3)+…+1/20(1+2+3+….+20)`
`=sum_(r=1)^(20)1/rcdot(r(r+1))/2`
`=(1+1)/2+(2+1)/2+(3+1)/2+…+(20+1)/2`
`=(20xx21)/4+10=115`