The value of `log_(5)5 + log_(5)5^(2) + log_(5)5^(3)`+…… + `log_(5)5^(n)`:

Options

A. n!
B. `n^(2)-1`
C. `((n+1)n)/2`
D. none of these

Answer

To solve the expression \( \log_{5}5 + \log_{5}5^{2} + \log_{5}5^{3} + \ldots + \log_{5}5^{n} \), we can follow these steps:

### Step 1: Rewrite the logarithmic terms
Each term in the series can be simplified using the property of logarithms:
\[ \log_{a}(b^{c}) = c \cdot \log_{a}(b) \]
Thus, we can rewrite the terms:
\[ \log_{5}5^{1} + \log_{5}5^{2} + \log_{5}5^{3} + \ldots + \log_{5}5^{n} = 1 \cdot \log_{5}5 + 2 \cdot \log_{5}5 + 3 \cdot \log_{5}5 + \ldots + n \cdot \log_{5}5 \]

### Step 2: Factor out the common logarithmic term
Since \( \log_{5}5 \) is common in all terms, we can factor it out:
\[ = \log_{5}5 \cdot (1 + 2 + 3 + \ldots + n) \]

### Step 3: Calculate the sum of the first n natural numbers
The sum of the first \( n \) natural numbers is given by the formula:
\[ 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]
Thus, we can substitute this sum into our expression:
\[ = \log_{5}5 \cdot \frac{n(n + 1)}{2} \]

### Step 4: Simplify using the value of \( \log_{5}5 \)
Since \( \log_{5}5 = 1 \), we can simplify further:
\[ = 1 \cdot \frac{n(n + 1)}{2} = \frac{n(n + 1)}{2} \]

### Final Answer
Thus, the value of \( \log_{5}5 + \log_{5}5^{2} + \log_{5}5^{3} + \ldots + \log_{5}5^{n} \) is:
\[ \frac{n(n + 1)}{2} \]

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