The sum `1+3+7+15+31+...to100` terms is `2^(100)-102 b` b. `2^(99)-101` c. `2^(101)-102` d. none of these

Options

A. `2^(100)-102`
B. `2^(99)-101`
C. `2^(101)-102`
D. none of these

Answer

C

Explanation

Here the successive differences are 2,4,8,16,… which are in G.P.
`S=1+3+7+15+31+…+T_(100)`
`=(2^(1)-1)+(2^(2)-1)+(2^(3)-1)+..+(2^(100)-1)`
`=(2+2^(2)+2^(3)+….+2^(100))-100`
`=2((2^(100)-1)/(2-1))-100`
`=2^(101)-102`