निम्नलिखित श्रेणी का nवां पद तथा n पदों का योगफल निकालिए ।
`1+(1-(1)/(2))+(1+(1)/(2)+(1)/(2^(2)))+.......`

Explanation

`t_(n)=1+(1)/(2)+(1)/(2^(2))+.......n` पदों तक
`=(1(1-(1)/(2^(n))))/(1-(1)/(2))=(1-(1)/(2^(n)))=2-(1)/(2^(n-1))`
अब `S_(n)=t_(1)+t_(2)+t_(3)+............+t_(n)`
`=(2-(1)/(2^(0)))+(2-(1)/(2)^(1))+(2-(1)/(2^(2)))+.........+(2-(1)/(2^(n-1)))`
`=(2+2+.......n` पदों तक `-((1)/(2^(0))+(1)/(2^(1))+(1)/(2^(2))+.......+(1)/(2^(n-1)))`
`=2n-(1-(1)/(2)+(1)/(2^(2))+.........+(1)/(2^(n-1)))`
`=2n-(1(-1(1)/(2^(n))))/(1-(1)/(2))=2n-2(1-(1)/(2^(n)))=2n-2+(1)/(2^(n-1))`