" The value of the expression "log_(2)(1+(1)/(2)sum_(k=1)^(11)^(12)C_(K))" is equal to "

sum_(x-1)^(20)k(1)/(2^(k)) is equal to

The Sum sum_(K=1)^(20)K(1)/(2^(K)) is equal to.

The value of the expression (sum_(r=0)^(10)C_(r))(sum_(k=0)^(10)(-1)^(k)(^^10C_(k))/(2^(k))) is :

The value of the expression (sum_(r=0)^10 "^10 C_r)(sum_(k=0)^10 (-1)^k (^10 C_k)/2^k) is :

The sum sum_(k=1)^(20) k (1)/(2^(k)) is equal to

The sum sum_(k=1)^(20) k (1)/(2^(k)) is equal to

The sum sum_(k=1)^(n)k(k^(2)+k+1) is equal to

For any integer K , let alpha_(k)=cos((kpi)/(7))+isin((kpi)/(7))," where " i=sqrt(-1) The value of the expression (sum_(k=1)^(12)|alpha_(k+1)-alpha_(k)|)/(sum_(k=1)^(3)|alpha_(4k-1)-alpha_(4k-2)|)

The value of the expression Sigma_(k=0)^(27)k.^(27)C_(k)((1)/(3))^(k)((2)/(3))^(27-k) is equal to