Find Σ_(r = 0)⁹ r · ⁹Cr and Σ_(r = 0)⁹ ⁹Cr/(r + 1).

The coefficients of $(1 + x)^n$ add up to $2^n$ (put $x = 1$), and the odd-placed and even-placed terms each add up to $2^{n - 1}$ (use $x = -1$). For a polynomial $f$, the even powers add up to $\dfrac{f(1) + f(-1)}{2}$ and the odd powers to $\dfrac{f(1) - f(-1)}{2}$. Also $\sum r \cdot {}^{n}C_{r} = n \cdot 2^{n - 1}$ and $\sum \dfrac{{}^{n}C_{r}}{r + 1} = \dfrac{2^{n + 1} - 1}{n + 1}$.