Sum to n terms the series 7+77+777+....`

Answer

To find the sum of the series \( S_n = 7 + 77 + 777 + \ldots \) up to \( n \) terms, we can express each term in a more manageable form.

### Step-by-Step Solution:

1. **Identify the Pattern**:
The terms of the series can be expressed as:
- 1st term: \( 7 \)
- 2nd term: \( 77 = 7 \times 11 \)
- 3rd term: \( 777 = 7 \times 111 \)
- 4th term: \( 7777 = 7 \times 1111 \)

We can see that each term can be represented as \( 7 \times (10^k - 1)/9 \) where \( k \) is the term index (starting from 0).

2. **Express Each Term**:
The \( k \)-th term can be written as:
\[ a_k = 7 \times \frac{10^k - 1}{9} \]
where \( k = 1, 2, \ldots, n \).

3. **Sum of the Series**:
The sum of the first \( n \) terms can be expressed as:
\[ S_n = \sum_{k=1}^{n} a_k = \sum_{k=1}^{n} 7 \times \frac{10^k - 1}{9} \]
This simplifies to:
\[ S_n = \frac{7}{9} \sum_{k=1}^{n} (10^k - 1) \]

4. **Separate the Sum**:
We can separate the sum:
\[ S_n = \frac{7}{9} \left( \sum_{k=1}^{n} 10^k - \sum_{k=1}^{n} 1 \right) \]
The second sum is simply \( n \).

5. **Geometric Series**:
The first sum \( \sum_{k=1}^{n} 10^k \) is a geometric series with first term \( 10 \) and common ratio \( 10 \):
\[ \sum_{k=1}^{n} 10^k = 10 \frac{10^n - 1}{10 - 1} = \frac{10^{n+1} - 10}{9} \]

6. **Combine the Results**:
Now substituting back:
\[ S_n = \frac{7}{9} \left( \frac{10^{n+1} - 10}{9} - n \right) \]
Simplifying gives:
\[ S_n = \frac{7}{81} (10^{n+1} - 10 - 9n) \]

### Final Answer:
The sum of the series \( 7 + 77 + 777 + \ldots \) up to \( n \) terms is:
\[ S_n = \frac{7}{81} (10^{n+1} - 10 - 9n) \]