If N, the set of natural numbers is partitioned into groups `S_(1)={1},S_(2)={2,3},S_(3)={4,5,6,7},S_(4)={8,9,10,11,12,13,14,15},"....",` find the sum of the numbers in `S_(50).`

Explanation

The numbers of terms in the groups are `1,2,2^(2),2^(3),"…."`
`therefore` The number of terms in the 50th group `=2^(50-1)=2^(49)`
`therefore` The first term of 1st group `=1=2^(0)=2^(1-1)`
The first term of 2nd group `=2=2^(1)=2^(2-1)`
The first term of 3rd group `=4=2^(2)=2^(3-1)`
`vdots" "vdots " " vdots " "vdots " " vdots "`
The first term of 50th group `2^(50-1)=2^(49)`
` therefore S_(50)=(2^(49))/(2){2xx2^(49)+(2^(49)-1)xx1}`
` =2^(48)(2^(50)+2^(49)-1)`
` =2^(48)[(2^(49)(2+1)-1]=2^(48)(3*2^(49)-1)`