Find the sum `1+(1+2)+(1+2+2^(2))+(1+2+2^(2)+2^(3))+` …. To n terms.

Explanation

nth term of the series,
`T_(n)=1+2+2^(2)+..+2^(n-1)`
`=(1(2^(n)-1))/(2-1)=2^(n)-1`
`therefore` Sum of the series,
`S=T_(1)+T_(2)+T_(3)+…T_(n)`
`=(2+2^(2)+2^(3)+….+2^(n))-(1+1+1+… n "times")`
`=(2(2^(n)-1))/(2-1)-n`
`=2^(n+1)-2-n`