Find the sum of n terms of the following series.
11 + 103 + 1005 + ........

Options

A. `10/9(10^n+1)+n^2`
B. `10/9(10^n-1)+n^2`
C. `10/9(10^n+1)+n`
D. `10/9(10^n-1)+n`

Answer

To find the sum of the first n terms of the series 11, 103, 1005, ..., we can break down the terms into two distinct series.

### Step 1: Identify the Series
The given series can be represented as:
- First term: 11 = 10 + 1
- Second term: 103 = 100 + 3
- Third term: 1005 = 1000 + 5

From this, we can see that the series can be split into two parts:
1. The series of the powers of 10: 10, 100, 1000, ...
2. The series of odd numbers: 1, 3, 5, ...

### Step 2: Analyze the First Series (Powers of 10)
The first series is:
- 10, 100, 1000, ...

This is a geometric progression (GP) where:
- First term (a) = 10
- Common ratio (r) = 10

The sum of the first n terms of a geometric series can be calculated using the formula:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \]

Substituting the values:
\[ S_n = \frac{10(10^n - 1)}{10 - 1} = \frac{10(10^n - 1)}{9} \]

### Step 3: Analyze the Second Series (Odd Numbers)
The second series is:
- 1, 3, 5, ...

This is an arithmetic progression (AP) where the sum of the first n odd numbers is given by:
\[ S = n^2 \]

### Step 4: Combine the Two Series
Now, we need to combine the sums of the two series to find the total sum of the first n terms of the original series:
\[ \text{Total Sum} = \text{Sum of GP} + \text{Sum of AP} \]
\[ \text{Total Sum} = \frac{10(10^n - 1)}{9} + n^2 \]

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