(6) (bonus) here is a proof that for n > 0, 12 22 +..+2" = 2n+1. proof. suppose 1 + 2 + 2 +..+ 2" = 2n+1 for some n > 0. then 2n+12n+1 .. +2" +2"n+1. 1222 using the inductive hypothesis 2(2"+1) 2"+2 2(n+1)+1 as we needed to show. now, obviously there is something wrong with this proof by induction since, for example, 1+222 7, but 22+1 = 23 = 8. what specifically is wrong with the proof?