It is clear that a(n)=2^(1-2^(-n)). In fact, for n=1 this produces 2^(1-1/2)=sqrt(2)=a1 and if it is true for a(n) then a(n+1) = sqrt (2 * 2^(1-2^(-n))) = sqrt(2^(2-2^(-n))) = 2^(1-2^(-(n+1))) (a) clearly 2^(1-2^(-n))<2<3 so the sequence is bounded by 3. Also a(n+1)/a(n) = 2^(1-2^(-n-1) - 1+2^(-n)) = 2^(1/2^n - 1/2^(n+1)) = 2^(1/2^(n+1)) >1 so the sequence is monotonically increasing. As it is monotonically increasing and has an upper bound it means it has a limin when n-> oo (b) 1-1/2^n -> 1 as n->oo so 2^(1-2^(-n)) -> 2 as n->oo
The action takes place in and around an old castle, sometimes
seemingly abandoned, sometimes occupied. The castle often contains secret passages,
trap doors, secret rooms, dark or hidden staircases, and possibly ruined sections. The
castle may be near or connected to caves, which lend their own haunting flavor with
their branches, claustrophobia, and mystery. (Translated into modern filmmaking, the
setting might be in an old house or mansion--or even a new house—where unusual
camera angles, sustained close ups during movement, and darkness or shadows create
the same sense of claustrophobia and entrapment.)
d. However
This demonstrates the contrast between the two sentences.
Its something in a poem that sounds the same in two different lines. Like sing and ring. Hope thus helps.
