If s n = r = 0 n 1 C r n and t n = r = 0 n r C r n , then t n s n is equal to

Question

If $s_{n} = \sum_{r = 0}^{n} \frac{1}{{}^{n}C_{r}}$ and $t_{n} = \sum_{r = 0}^{n} \frac{r}{{}^{n}C_{r}},$ then $\frac{t_{n}}{s_{n}}$ is equal to

Smart Achievers · Accepted Answer

(3)

$t_{n} = \sum_{r = 0}^{n} \frac{r}{{}^{n}C_{r}} = \sum_{r = 0}^{n} \frac{n - (n - r)}{{}^{n}C_{n - r}}$

$\Rightarrow t_{n} = n \sum_{r = 0}^{n} \frac{1}{{}^{n}C_{r}} - \sum_{r = 0}^{n} \frac{n - r}{{}^{n}C_{n - r}}$

$t_{n} = n \sum_{r = 0}^{n} \frac{1}{{}^{n}C_{r}} - \sum_{r = 0}^{n} \frac{r}{{}^{n}C_{r}}$ $[Replacing n - r by r]$

$t_{n} = n s_{n} - t_{n}$ $∴$ $\frac{t_{n}}{s_{n}} = \frac{n}{2}$

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