Differentiate (1+x)^n = sum C(n,r)x^r: n(1+x)^{n-1} = sum rC(n,r)x^{r-1}. Put x = 1: n2^{n-1} = sum rC(n,r). This gives the sum of rC(n,r).

Multiply by x first, then differentiate: x*(1+x)^n = sum C(n,r)*x^{r+1}. Differentiating: (1+x)^n + nx(1+x)^{n-1} = sum (r+1)*C(n,r)*x^r.

For sum r^2*C(n,r): use the identity r^2 = r(r-1) + r. Then sum r(r-1)C(n,r) = n(n-1)2^{n-2}, and sum rC(n,r) = n2^{n-1}. Total: n(n-1)2^{n-2} + n2^{n-1} = n(n+1)*2^{n-2}.

Integrate (1+x)^n from 0 to x: (1+x)^{n+1}/(n+1) - 1/(n+1) = sum C(n,r)*x^{r+1}/(r+1). Put x = 1: (2^{n+1} - 1)/(n+1) = sum C(n,r)/(r+1).