In a`DeltaABC`, prove that
`underset(r=0)overset(n)Sigma .^(n)C_(r)a^(r)b^(n-r) cos ( r B-(n-r)A)=c^(n)`.

In a DeltaABC , prove that Sigma_(r=0)^(n)""^(n)C_(r)a^(r)b^(n-r) cos ( r B-(n-r)A)=c^(n) .

Prove that overset(n)underset(r=0)(Sigma^(n))C_(r).4^(r)=5^(n)

Prove that overset(n)underset(r=0)(Sigma^(n))C_(r).4^(r)=5^(n)

underset(r=1)overset(n)(sum)r(.^(n)C_(r)-.^(n)C_(r-1)) is equal to

Prove that Sigma_(r=1) ""^(n)C_(r).3^(r)=4^(n)

If x + y = 1 , prove that sum_(r=0)^(n) r""^(n)C_(r) x^(r ) y^(n-r) = nx .

If n is a positive integer, prove that underset(r=1)overset(n)sumr^(3)((""^(n)C_(r))/(""^(n)C_(r-1)))^(2)=((n)(n+1)^(2)(n+2))/(12)

Prove that Sigma_(r=0)^(n) ""^(n)C_(r).3^(r)=4^(n)

Prove that sum_(r=0)^(n) ""^(n)C_(r).(n-r)cos((2rpi)/(n)) = - n.2^(n-1).cos^(n)'(pi)/(n) .

Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2.""^(4n-1)C_(2n-1) .