Find the sum to n terms of the series
11+ 102+1003+10004+… :

Options

A. a)`(10^n-1) + (n(n+1))/(2)`
B. b)`(10)/(9) (10^n -1) + (n(n+1))/(2)`
C. `c)10^n + n^2 -1`
D. d)none of these

Answer

To find the sum to n terms of the series \( 11 + 102 + 1003 + 10004 + \ldots \), we can break down the terms of the series and identify a pattern.

### Step-by-Step Solution:

1. **Identify the Pattern in the Series:**
The terms of the series can be expressed as:
- \( 11 = 10^1 + 1 \)
- \( 102 = 10^2 + 2 \)
- \( 1003 = 10^3 + 3 \)
- \( 10004 = 10^4 + 4 \)

We can see that the \( n \)-th term can be expressed as:
\[ T_n = 10^n + n \]

2. **Write the Sum of the First n Terms:**
The sum \( S_n \) of the first \( n \) terms can be written as:
\[ S_n = T_1 + T_2 + T_3 + \ldots + T_n = (10^1 + 1) + (10^2 + 2) + (10^3 + 3) + \ldots + (10^n + n) \]

3. **Separate the Series:**
We can separate the sum into two parts:
\[ S_n = (10^1 + 10^2 + 10^3 + \ldots + 10^n) + (1 + 2 + 3 + \ldots + n) \]

4. **Calculate the Geometric Series:**
The first part \( 10^1 + 10^2 + 10^3 + \ldots + 10^n \) is a geometric series where:
- First term \( a = 10 \)
- Common ratio \( r = 10 \)
- Number of terms \( n \)

The sum of a geometric series can be calculated using the formula:
\[ S = a \frac{r^n - 1}{r - 1} \]
Thus,
\[ S_1 = 10 \frac{10^n - 1}{10 - 1} = \frac{10}{9} (10^n - 1) \]

5. **Calculate the Arithmetic Series:**
The second part \( 1 + 2 + 3 + \ldots + n \) is an arithmetic series, and its sum can be calculated using the formula:
\[ S_2 = \frac{n(n + 1)}{2} \]

6. **Combine Both Parts:**
Now, we can combine both parts to find \( S_n \):
\[ S_n = \frac{10}{9} (10^n - 1) + \frac{n(n + 1)}{2} \]

### Final Expression for the Sum:
Thus, the sum of the first \( n \) terms of the series is:
\[ S_n = \frac{10}{9} (10^n - 1) + \frac{n(n + 1)}{2} \]