If `S_(n)` denotes the sum to `n` terms of the series `(1 le n le 9)1 + 22 + 333 + "……" + ("nnn…..n")/(n"times")`, then for `n ge 2`

Options

A. `S_(n) - S_(n - 1) = (1)/(9)(10^(n) - n^(2) + n)`
B. `S_(n) = (1)/(9)(10^(n) - n^(2) + 2n - 2)`
C. `9(S_(n) - S_(n-1)) = n(10^(n) - 1)`
D. `S_(3) = 356`

Explanation

To solve the problem, we need to find the sum \( S_n \) of the series defined as:

\[ S_n = 1 + 22 + 333 + \ldots + \underbrace{nnn\ldots n}_{n \text{ times}} \]

for \( n \geq 2 \).

### Step 1: Express the terms of the series
The terms in the series can be expressed as:
- The first term is \( 1 \)
- The second term is \( 22 = 2 \times 11 \)
- The third term is \( 333 = 3 \times 111 \)
- The \( n \)-th term is \( n \) repeated \( n \) times, which can be expressed as \( n \times \underbrace{111\ldots1}_{n \text{ times}} \).

The term \( 111\ldots1 \) (with \( n \) ones) can be represented as:
\[ \frac{10^n - 1}{9} \]
Thus, the \( n \)-th term can be expressed as:
\[ n \times \frac{10^n - 1}{9} \]

### Step 2: Write the sum \( S_n \)
Now, we can write the sum \( S_n \) as:
\[ S_n = 1 + 2 \times \frac{10^2 - 1}{9} + 3 \times \frac{10^3 - 1}{9} + \ldots + n \times \frac{10^n - 1}{9} \]

### Step 3: Factor out \( \frac{1}{9} \)
Factoring out \( \frac{1}{9} \) from the sum:
\[ S_n = \frac{1}{9} \left( 1 + 2(10^2 - 1) + 3(10^3 - 1) + \ldots + n(10^n - 1) \right) \]

### Step 4: Simplify the expression
We can separate the terms:
\[ S_n = \frac{1}{9} \left( \sum_{k=1}^{n} k \cdot 10^k - \sum_{k=1}^{n} k \right) \]
The second sum \( \sum_{k=1}^{n} k \) can be calculated using the formula:
\[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \]

### Step 5: Calculate the first sum
The first sum \( \sum_{k=1}^{n} k \cdot 10^k \) can be calculated using the formula for the sum of a geometric series. The sum can be derived as:
\[ \sum_{k=1}^{n} kx^k = x \frac{d}{dx} \left( \frac{x(1-x^n)}{1-x} \right) \]
Setting \( x = 10 \) gives us the required sum.

### Step 6: Combine results
After calculating both sums, we substitute back into the expression for \( S_n \) and simplify.

### Final Expression
The final expression for \( S_n \) will be:
\[ S_n = \frac{1}{9} \left( \text{(result of } \sum_{k=1}^{n} k \cdot 10^k \text{)} - \frac{n(n+1)}{2} \right) \]