If `(1+x)^n=sum_(r=0)^n C_r x^r ,` then prove that `C_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot` .

If (1+x)^n=sum_(r=0)^n^n C_r , show that C_0+(C_1)/2++(C_n)/(n+1)=(2^(n+1)-1)/(n+1) .

If (1+2x+x^2)^n=sum_(r=0)^(2n)a_r x^r ,then a_r is a. (.^nC_2)^2 b. .^n C_r .^n C_(r+1) c. .^(2n) C_r d. .^(2n) C_(r+1)

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^2)^n=sum_(r=0)^n a_r x^r(1-x)^(2n-r),t h e na_r is equal to ^n C_r b. ^n C_r3^r c. ^2n C_r d. ^n C_r2^r

Let (1 + x)^(n) = sum_(r=0)^(n) C_(r) x^(r) and , (C_(1))/(C_(0)) + 2 (C_(2))/(C_(1)) + (C_(3))/(C_(2)) +…+ n (C_(n))/(C_(n-1)) = (1)/(k) n(n+1) , then the value of k, is

If (1+x)^(n)=C_(0)+C_(1).x+C_(2).x^(2)+C_(3).x^(3)+......+C_(n).x^(n), then prove that C_(0)+2C_(1)+4C_(2)+6C_(3)+...+2n.C_(n)=1+n*2^(n)

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

Prove that "^n C_r+^(n-1)C_r+...+^r C_r=^(n+1)C_(r+1) .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that (1*2) C_(2) + (2*3) C_(3) + …+ {(n-1)*n} C_(n) = n(n-1) 2^(n-2) .