If `T_(r)=(1)/(log_(z)4)` (where `r in N`), then the value of `sum_(r=1)^(4) T_(r)` is :

Options

A. 3
B. 4
C. 5
D. 10

Answer

To solve the problem, we need to find the value of \( \sum_{r=1}^{4} T_r \) where \( T_r = \frac{1}{\log_z 4} \).

### Step 1: Understand the expression for \( T_r \)
We know that \( T_r = \frac{1}{\log_z 4} \). The base \( z \) is not specified in the problem, but we can express \( \log_z 4 \) in terms of logarithms with base 2.

### Step 2: Rewrite \( \log_z 4 \)
Using the change of base formula, we have:
\[ \log_z 4 = \frac{\log_2 4}{\log_2 z} \]
Since \( 4 = 2^2 \), we can simplify:
\[ \log_2 4 = 2 \]
Thus,
\[ \log_z 4 = \frac{2}{\log_2 z} \]

### Step 3: Substitute back into \( T_r \)
Now substituting this back into the expression for \( T_r \):
\[ T_r = \frac{1}{\log_z 4} = \frac{\log_2 z}{2} \]

### Step 4: Calculate \( \sum_{r=1}^{4} T_r \)
Now we can compute the sum:
\[ \sum_{r=1}^{4} T_r = T_1 + T_2 + T_3 + T_4 = 4 \cdot \frac{\log_2 z}{2} \]
This simplifies to:
\[ \sum_{r=1}^{4} T_r = 2 \log_2 z \]

### Step 5: Determine \( z \)
Since the problem does not specify \( z \), we can assume \( z = 2 \) (as it is common in logarithmic problems). Thus:
\[ \log_2 2 = 1 \]
So,
\[ \sum_{r=1}^{4} T_r = 2 \cdot 1 = 2 \]

### Final Answer
The value of \( \sum_{r=1}^{4} T_r \) is:
\[ \boxed{2} \]