Prove that `C_1/C_0+(2c_(2))/C_1+(3C_3)/(C_2)+......+(n.C_n)/(C_(n-1))=(n(n+1))/2`
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With usual notations prove that C_1/C_0 + 2. C_2/C_1 + 3.C_3/C_2 + ……+n.(C_n)/(C_(n-1)) = (n(n +1))/(2) Hence prove that (15C_1)/(15C_0) + 2.(15C_2)/(15C_1) + 3. (15C_3)/(15C_2) +……..+ 15. (15C_15)/(15C_14) = 120

C_1/C_0+2C_2/C_1+3C_3/C_2+............+nC_n/C_(n-1)=(n(n+1))/2

If (1+x)^(n)=C_(0)+C_(1)x+…..+C_(n)x^(n) , then (C_(1))/(C_(0))+(2C_(2))/(C_(1))+(3C_(3))/(C_(2))+....+(nC_(n))/(C_(n-1)) is :

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

Prove that (C_0+C_1)(C_1+C_2)(C_2+C_3)(C_3+C_4)...........(C_(n-1)+C_n) = (C_0C_1C_2.....C_(n-1)(n+1)^n)/(n!)

Prove that nC_1(nC_2)^2(n C_3)^3.......(n C_n)^n le ((2^n)/(n+1))^((n+1)C_2),AAn in Ndot

Prove that C_(1)^(2)-2*C_(2)^(2)+3*C_(3)^(2)-…-2n*C_(2n)^(2)=(-1)^(n)n*C_(n)

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) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - (C_(1))/(2) + (C_(2))/(3) -…+ (-1)^(n) (C_(n))/(n+1) = (1)/(n+1) .

Prove that: .^(2)C_(2)+^(3)C_(2)+^(4)C_(2)+…..+^(n+1)C_(2)=1/6n(n+1)(n+2)