`log_(10)10+log_(10)10^(2)+ . . .+log_(10)10^(n)`

Options

A. A)`n^(2)+1`
B. B)`n^(2)-1`
C. C)`((n^(2)+n)/(3))`
D. D)`(n^(2)+n)/(2)`

Answer

To solve the expression \( \log_{10} 10 + \log_{10} 10^2 + \ldots + \log_{10} 10^n \), we can follow these steps:

### Step 1: Simplify Each Logarithmic Term
Using the property of logarithms that states \( \log_b (a^c) = c \cdot \log_b a \), we can simplify each term in the series:

- The first term is \( \log_{10} 10 = 1 \) (since \( \log_a a = 1 \)).
- The second term is \( \log_{10} 10^2 = 2 \cdot \log_{10} 10 = 2 \).
- The third term is \( \log_{10} 10^3 = 3 \cdot \log_{10} 10 = 3 \).
- Continuing this pattern, the \( k \)-th term is \( \log_{10} 10^k = k \).

Thus, the series can be rewritten as:
\[ 1 + 2 + 3 + \ldots + n \]

### Step 2: Use the Formula for the Sum of the First n Natural Numbers
The sum of the first \( n \) natural numbers can be calculated using the formula:
\[ S_n = \frac{n(n + 1)}{2} \]

### Step 3: Substitute into the Formula
Substituting \( n \) into the formula gives us:
\[ S_n = \frac{n(n + 1)}{2} \]

### Conclusion
Thus, the value of the expression \( \log_{10} 10 + \log_{10} 10^2 + \ldots + \log_{10} 10^n \) simplifies to:
\[ \frac{n(n + 1)}{2} \]

### Final Answer
The final answer is:
\[ \frac{n(n + 1)}{2} \]

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