`C_0+2. C_1+3. C_2+............+(n+1)*C_n=`

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : C_0-2C_1+3C_2-.........+(-1)^n (n+1)C_n=0 .

C_ (0) ^ (2) + 2C_ (1) ^ (2) + 3.C_ (2) ^ (2) + ............ + (n + 1) C_ (n ) ^ (2) =

Find the sum of C_0 + 2C_1 + 3C_2 + .... + (n+1)C_n

Show that C_0 + 2C_1 + 3C_2 +4C_3 +...+ (n + 1)C_n = (n + 2)2^(n-1)

(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!)

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

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

Show 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