If `(1+x)^n=sum_(r=0)^n C_r x^r ,` then prove that `C_1+2c_2+3C_1++n C_n=n2^(n-1)dot` .

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r) then prove that C_(1)+2C_(2)+3C_(3)+....+nC_(n)=n2^(n-1)

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r) then prove that C_(0)+2C_(1)+3C_(2)+.....+(n+1)C_(n)=2^(n-1)(n+2)

If (1+x)^n=sum_(r=0)^n C_rx^r then prove that sum_(r=0)^n (C_r)/((r+1)2^(r+1))=(3^(n+1)-2^(n+1))/((n+1)2^(n+1))

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r) then prove that C_(0)+(C_(1))/(2)+......+(C_(n))/(n+1)=2

If (1+x)^n=underset(r=0)overset(n)C_(r)x^r then prove that C_(1)^2+2.C_(2)^(2)+3.C_(3)^2 +…….+n.C_(n)^(2)=((2n-1)!/((n-1)!)^2

If (1 + x)^(n) = sum_(r=0)^(n) C_(r),x^(r) , then prove that (sumsum)_(0leiltjlen) ((i)/(C_(i)) + (j)/(C_(j))) = (n^(2))/(2) sum_(r=0)^(n) (1)/(C_(r)) .

If (1 + x)^(n) = sum_(r=0)^(n) C_(r) x^(r),(1 + (C_(1))/(C_(0))) (1 + (C_(2))/(C_(1)))...(1 + (C_(n))/(C_(n-1))) is equal to

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r), show that (C_(0))/(2)+(C_(1))/(3)+(C_(2))/(4)+...+(C_(n))/(n+2)=(n*2^(n+1)+1)/((n+1)(n+2))