The value of
`"log"_(5) (1+ (1)/(5)) + "log"_(5) (1+(1)/(6)) + "log"_(5)(1+(1)/(7)) + …. + "log"_(5)(1+(1)/(624))` is

Options

A. 5
B. 4
C. 3
D. 2

Answer

To solve the problem, we need to evaluate the expression:

\[ \text{log}_{5} \left(1 + \frac{1}{5}\right) + \text{log}_{5} \left(1 + \frac{1}{6}\right) + \text{log}_{5} \left(1 + \frac{1}{7}\right) + \ldots + \text{log}_{5} \left(1 + \frac{1}{624}\right) \]

### Step 1: Rewrite Each Term
Each term in the sum can be rewritten as follows:

\[ \text{log}_{5} \left(1 + \frac{1}{n}\right) = \text{log}_{5} \left(\frac{n + 1}{n}\right) = \text{log}_{5} (n + 1) - \text{log}_{5} (n) \]

Thus, we can rewrite the entire expression as:

\[ \sum_{n=5}^{624} \left( \text{log}_{5} (n + 1) - \text{log}_{5} (n) \right) \]

### Step 2: Recognize the Telescoping Series
The above sum is a telescoping series. When we expand it, we see that most terms will cancel out:

\[ \left( \text{log}_{5} (6) - \text{log}_{5} (5) \right) + \left( \text{log}_{5} (7) - \text{log}_{5} (6) \right) + \ldots + \left( \text{log}_{5} (625) - \text{log}_{5} (624) \right) \]

### Step 3: Simplify the Series
After cancellation, we are left with:

\[ \text{log}_{5} (625) - \text{log}_{5} (5) \]

### Step 4: Use Logarithmic Properties
Using the property of logarithms that states \(\text{log}_{a}(b) - \text{log}_{a}(c) = \text{log}_{a}\left(\frac{b}{c}\right)\), we can combine the logs:

\[ \text{log}_{5} \left(\frac{625}{5}\right) \]

### Step 5: Calculate the Value
Now, calculate \(\frac{625}{5}\):

\[ \frac{625}{5} = 125 \]

Thus, we have:

\[ \text{log}_{5} (125) \]

### Step 6: Final Calculation
Since \(125 = 5^3\), we can simplify further:

\[ \text{log}_{5} (125) = \text{log}_{5} (5^3) = 3 \]

### Final Answer
The value of the original expression is:

\[ \boxed{3} \]