Find the sum of the series `1+2(1-x)+3(1-x)(1-2x)+....+n(1-x)(1-2x) (1-3x)............[1-(n-1)x].

Answer

`1/x[1-(1-x)(1-2x)...(1-nx)`

Explanation

`T_(r)=r(1-x)(1-2x)(1-3x)…(1-(r-1)x)`
`=(1-x)(1-2x)(1-3x)…(1-(r-1)x)(xr-1+1)/x`
`=1/x{(1-x)(1-2x)…(1-(r-1)x)-(1-x)(1-2x)..(1-(r-1)x)(1-rx)}`
`=1/x[V(r-1)-V(r )]`, where
`V(r )=(1-x)(1-2x)…(1-rx)`
`therefore` Required sum=`sum_(r=1)^(n)T_(r)=1/xsum_(r=1)^(n)V(r-1)-V( r)`
`=1/x(V(0)-V(n))`
`1/x[1-(1-x)(1-2x)...(1-nx)]`