Question

please help me with this question asap!​

Answer 1

$\sf {(1 + x)}^{n} = C_{0} + xC_{1} + ... + {x}^{n } C_{n}$

$\sf x {(1 + x)}^{n} =x C_{0} + x^{2} C_{1} + ... + {x}^{n + 1 } C_{n}$

Differentiating both the sides w.r.t. x

$\\ \sf nx {(1 + x)}^{n - 1} + (1 + x) ^{n} = \sum _{r = 0} ^{n} (r + 1) ^{n} C_{r} {x}^{r}$

Substituting x=1

$\\ \sf C_{0} + 2C_{1} + ... + (n + 1)C _{n} = n {2}^{n - 1} + {2}^{n}$

$\\ \boxed{\sf \sum _{r = 0} ^{n} (r + 1) ^{n} C_{r} {x}^{r} = (n + 2) {2}^{n - 1}}$