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The image you uploaded contains a mathematical problem related to summing the terms of a series. Let's break down the information provided and solve the problem.

### Problem Summary:
- The problem involves finding the sum of terms in a series.
- The series provided is: \( 3 + 33 + 333 + 3333 + \dots \) up to \( n \) terms.
- The general term for the series appears to involve numbers like \( 3, 33, 333, 3333, \dots \).

### General Term Identification:
Let's identify the general term in the sequence:

Each term seems to follow a pattern:
- 1st term: \( 3 = 3 \times 1 \)
- 2nd term: \( 33 = 3 \times 11 \)
- 3rd term: \( 333 = 3 \times 111 \)
- 4th term: \( 3333 = 3 \times 1111 \)

This suggests that the \( n \)-th term in the series is:
\[ T_n = 3 \times \underbrace{111\ldots1}_{n \text{ digits}} \]

The number \( \underbrace{111\ldots1}_{n \text{ digits}} \) can be expressed as:
\[ \underbrace{111\ldots1}_{n \text{ digits}} = \frac{10^n - 1}{9} \]

Thus, the \( n \)-th term in the series becomes:
\[ T_n = 3 \times \frac{10^n - 1}{9} = \frac{3 \times (10^n - 1)}{9} = \frac{10^n - 1}{3} \]

### Sum of the Series:
The sum of the series up to \( n \) terms \( S_n \) is given by:
\[ S_n = \sum_{k=1}^{n} T_k = \sum_{k=1}^{n} \frac{10^k - 1}{3} \]

This can be simplified as:
\[ S_n = \frac{1}{3} \sum_{k=1}^{n} (10^k - 1) \]
\[ S_n = \frac{1}{3} \left( \sum_{k=1}^{n} 10^k - \sum_{k=1}^{n} 1 \right) \]

We can now evaluate each sum separately:

1. **Sum of the geometric series \( 10^k \) from \( k=1 \) to \( n \):**
   \[ \sum_{k=1}^{n} 10^k = 10 \times \frac{10^n - 1}{9} \]

2. **Sum of constants \( \sum_{k=1}^{n} 1 = n \):**

Substituting these into the sum \( S_n \):
\[ S_n = \frac{1}{3} \left( \frac{10(10^n - 1)}{9} - n \right) \]
\[ S_n = \frac{10(10^n - 1) - 9n}{27} \]

This expression gives the sum of the series up to \( n \) terms.

Would you like a detailed explanation of any part of the solution, or do you have any other questions related to this problem?

### Relative Questions:
1. How do you derive the formula for the general term of a series?
2. What are the key steps to sum a geometric series?
3. Can you generalize the sum formula for any series of similar type?
4. How does the formula change if the series terms have a different base (e.g., \( 2, 22, 222, \dots \))?
5. What is the significance of the factor \( \frac{1}{3} \) in the sum formula?

### Tip:
When dealing with series, identifying the pattern in the general term is crucial. Once you have the general term, summing the series often involves recognizing it as a geometric or arithmetic progression.

Sum the first n terms of the series 3 + 33 + 333 + 3333 + ...

Explore the solution to finding the sum of the series 3 + 33 + 333 + ... up to n terms. This problem involves concepts from geometric and arithmetic series, providing a step-by-step breakdown of the formulas used.

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