If `s_(n)=sum_(r=0)^(n)(1)/(.^(n)C_(r))and t_(n)=sum_(r=0)^(n)(r)/(.^(n)C_(r))`, then `(t_(n))/(s_(n))` is equal to

Options

A. `n-1`
B. `(1)/(2)n-1`
C. `(1)/(2)n`
D. `(2n-1)/(2)`

Answer

To solve the problem, we need to evaluate the expressions for \( s_n \) and \( t_n \) and then find the ratio \( \frac{t_n}{s_n} \).

### Step 1: Define \( s_n \)
The expression for \( s_n \) is given by:
\[ s_n = \sum_{r=0}^{n} \frac{1}{\binom{n}{r}} \]
This represents the sum of the reciprocals of the binomial coefficients.

### Step 2: Define \( t_n \)
The expression for \( t_n \) is given by:
\[ t_n = \sum_{r=0}^{n} \frac{r}{\binom{n}{r}} \]
This represents the sum of the terms where each term is the index \( r \) multiplied by the reciprocal of the binomial coefficient.

### Step 3: Simplify \( t_n \)
We can rewrite \( t_n \) as follows:
\[ t_n = \sum_{r=0}^{n} \frac{r}{\binom{n}{r}} = \sum_{r=1}^{n} \frac{r}{\binom{n}{r}} = \sum_{r=1}^{n} \frac{n}{n} \cdot \frac{r}{\binom{n}{r}} = n \sum_{r=1}^{n} \frac{1}{\binom{n-1}{r-1}} \]
This is because \( r \cdot \binom{n}{r} = n \cdot \binom{n-1}{r-1} \).

### Step 4: Relate \( t_n \) to \( s_n \)
Now, we can express \( t_n \) in terms of \( s_n \):
\[ t_n = n \sum_{r=0}^{n-1} \frac{1}{\binom{n-1}{r}} = n s_{n-1} \]
where \( s_{n-1} = \sum_{r=0}^{n-1} \frac{1}{\binom{n-1}{r}} \).

### Step 5: Find the ratio \( \frac{t_n}{s_n} \)
Now we can find the ratio:
\[ \frac{t_n}{s_n} = \frac{n s_{n-1}}{s_n} \]

### Step 6: Evaluate \( s_n \)
From the properties of binomial coefficients, we know that:
\[ s_n = \sum_{r=0}^{n} \frac{1}{\binom{n}{r}} = \frac{2^n}{\binom{n}{0}} = 2^n \]
Thus, we can write:
\[ s_n = 2^n \]

### Step 7: Substitute \( s_n \) into the ratio
Now substituting \( s_n \) into the ratio gives:
\[ \frac{t_n}{s_n} = \frac{n s_{n-1}}{2^n} \]

### Step 8: Evaluate \( s_{n-1} \)
Similarly, we can evaluate \( s_{n-1} \):
\[ s_{n-1} = 2^{n-1} \]
So, substituting this into our ratio:
\[ \frac{t_n}{s_n} = \frac{n \cdot 2^{n-1}}{2^n} = \frac{n}{2} \]

### Final Answer
Thus, we conclude that:
\[ \frac{t_n}{s_n} = \frac{n}{2} \]