AP EAMCET201822 Apr 2018Evening ShiftMathematicsLimitsActual
aligned & _ n n^ -n k & (n+1) (n+ 1 2 ) (n+ 1 2^2 ) (n+ 1 2^ k-1 ) ^n= aligned
Options
- A2
- Be^ 2 (1- 1 2^k )
- C2 (1- 1 2^k )
- De^2
Correct answer
B. e^ 2 (1- 1 2^k )
Step-by-step solution
aligned & _ n n^ -n k (n+1) (n+ 1 2 ) (n+ 1 2^2 ) (n+ 1 2^ k-1 ) ^n & =P Let & P= _ n ( .(n+1) (n+ 1 2^1 ) (n+ 1 2^2 ) (n+ 1 2^ k-1 ) )^n n^k ) aligned = _ n [ ( n+ 1 2^0 n ) ( n+ 1 2^1 n ) ( n+ 1 2^2 n ) ( n+ 1 2^ k-1 n ) ]^n So, aligned & P= _ n n [ array l (1+ 1 2^0 n )+ (1+ 1 2^1 n ) + (1+ 1 2^2 n )+ + (1+ 1 2^ k-1 n ) array ] & = _ n [ array c . (1+ 1 2^ n ) 1 / n + (1+ 1 2^1 n ) 1 / n + (1+ 1 2^2 n ) 1 / n ] + + (1+ 1 2^ k-1 n ) 1 / n array ] & = 1 2^ + 1 2^1 + 1 2^2 + .+ 1 2^ k-1 = 1- 1 2^k 1- 1 2 =2 (1- 1 2