The sum of the series (1+2)+(1+2+22)+(1+2+22+23)+.... upto n terms is

The given series is(1+2)+(1+2+22)+(1+2+22+23)+.....+n terms ∴     Tn=1+2+22+23+...+2n  =1(2n+1−1)2−1=2n+1−1∑1nTn=∑1n(2n+1−1)=∑1n2n+1−∑1n1                                =(22+23+ ... +2n+1)−n                                =2(2+22+ .. +2n)−n                                =4(2n−1)2−1−n=2n+2−n−4