Prove that `sum_(r=0)^n 3^r ` `nCundersetr` = `4^n`.

Prove that sum_(r=0)^n^n C_rsinr xcos(n-r)x=2^(n-1)sin(n x)dot

sum_(r=0)^(m)( n+r )C_(n)=

Evaluate sum_(r=1)^(n)rxxr!

The natural numbers arearranged innthe form given below The rth group containing 2^(r-1) numbers. Prove that sum of the numbers in the nth group is 2^(n-2)[2^(n)+2^(n-1)-1] .

Prove by mathematical induction that sum_(r=0)^(n)r^(n)C_(r)=n.2^(n-1), forall n in N .

Find he value of sum_(r=1)^(4n+7)\ i^r where, i=sqrt(- 1).

lim_(n rarr oo) sum_(r=0)^(n-1) 1/(n+r) =

Prove that the relation R defined on the set N of natural numbers by xRy iff 2x^(2) - 3xy + y^(2) = 0 is not symmetric but it is reflexive.

If a_n=sum_(r=0)^n1/(""^nC_r) , then sum_(r=0)^nr/(""^nC_r) equals :

Let n be an odd integer. If sin n theta=sum_(r=0)^(n)b_(r) sin^(r)theta for every value of theta , then