$1+2x+3x^3+4x^2+......+\left(r+1\right)x^r+.......\infty\ ধারাটির\ যোগফল\ কত?$

Find the sum of the series $\displaystyle\sum _{ r=0 }^{ n }{ { \left( -1 \right) }^{ n } } { _{ }^{ n }{ C } }_{ r }\left[ \cfrac { 1 }{ { 2 }^{ r } } +\cfrac { { 3 }^{ r } }{ { 2 }^{ 2r } } +\cfrac { { 7 }^{ r } }{ { 2 }^{ 3r } } +\cfrac { { 15 }^{ r } }{ { 2 }^{ 4r } } +...upto\: m\: terms \right]$

Let n be a positive integer.  Then the value of   $\sum_{k = 0}^{n} \left ( - 1 \right )^{k} \left ( \frac{n}{k} \right )$    is -- 

The arithmetic mean of $^nC_0 , \ ^nC_1, \ ^nC_2 ...., \ ^nC_n$ is ;

Number of different terms in the sum $( 1 + x ) ^ { 2009 } \cdot \left( 1 + x ^ { 2 } \right) ^ { 2008 } + \left( 1 + x ^ { 3 } \right) ^ { 2007 } ,$ is