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$T(n) =$ if $n=1$, then time execution is $1$, if $n \geq 2$ then $2T(n-1)+n$

The options are:

  1. $T(n) = 2^{n+1} - n - 2$
  2. $T(n) = O(n2^n)$
  3. $T(n) = \Omega(n)$
  4. $T(n) = \theta(2^n)$

Thanks.

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3  
What have you tried? –  lhf May 11 '12 at 0:12
    
The first choice seems satisfying what the properties you listed exactly. –  Shuhao Cao May 11 '12 at 0:12
    
I've removed the "complex-analysis" tag since this isn't related to complex analysis. –  Antonio Vargas May 11 '12 at 0:37
2  
Define $T(n) = f(n) -n -2$. And see if you can get $f(n)$ easily. –  user17762 May 11 '12 at 3:42
    
The title is wrong; you're not looking for the complexity of this function; you're looking for a closed form for it (option 1) or its asymptotic behaviour (options 2 to 4). This is probably related to the time complexity of a an algorithm, but complexity is a property of the algorithm, not of the function specifying its execution time. –  joriki May 11 '12 at 6:57

1 Answer 1

Ignoring the choices you have for the time being, here is how we could start out.

Define $S(n)=2^{-n}T(n)$, then, after multiplying by $2^{-n}$, the recurrence becomes $$ S(n)=S(n-1)+\frac{n}{2^n}\tag{1} $$ Equation $(1)$ tells us that $S(n)$ is the finite sum $$ c+\sum_{k=1}^n\frac{k}{2^k}\tag{2} $$ where $c$ is a constant chosen so that $S(1)=\frac12$.

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