S1,S2,S3,...Sn are sums of n infinite geometric progressions. The first terms of these progressions are 1,22-1,23-1,...2n-1 and the common ratios are 1/2,1/22,1/23...1/2n. Calculate the value of S1+S2+...+Sn

S1 = 1 /(1-1/2) = 2
S2=22 -11-122 =22 S3 =23-11-123 =23 ..Sn =2nSo S1 + S2 + S3 ..Sn = 2 + 22 + 23 + 24 ..+2n =2(2n-12-1)=2n+1-2 (ans)

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