Find the sum of n terms of the series`" 7 + 77 + 777 + …`

Explanation

To find the sum of the first n terms of the series \(7 + 77 + 777 + \ldots\), we can break down the problem step by step.

### Step 1: Identify the pattern in the series
The series consists of terms that can be expressed in a specific form. The first term is 7, the second term is 77, and the third term is 777. We can express these terms as:
- \(7 = 7\)
- \(77 = 7 \times 11\)
- \(777 = 7 \times 111\)

We can see that each term can be represented as \(7\) multiplied by a number that consists of repeating digits of \(1\).

### Step 2: Express the terms in a more manageable form
The \(n\)-th term can be expressed as:
\[ T_n = 7 \times (10^n - 1)/9 \]
This is derived from the fact that \(11, 111, 1111, \ldots\) can be represented as:
\[ \frac{10^n - 1}{9} \]
Thus, the series can be rewritten as:
\[ S_n = 7 \left( \frac{10^1 - 1}{9} + \frac{10^2 - 1}{9} + \frac{10^3 - 1}{9} + \ldots + \frac{10^n - 1}{9} \right) \]

### Step 3: Factor out the common terms
We can factor out \(7/9\) from the sum:
\[ S_n = \frac{7}{9} \left( (10 + 100 + 1000 + \ldots + 10^n) - n \right) \]

### Step 4: Identify the sum of the geometric series
The sum \(10 + 100 + 1000 + \ldots + 10^n\) 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:
\[ S = a \frac{r^n - 1}{r - 1} \]
Substituting the values:
\[ S = 10 \frac{10^n - 1}{10 - 1} = \frac{10}{9} (10^n - 1) \]

### Step 5: Substitute back into the sum expression
Now, substituting this back into our expression for \(S_n\):
\[ S_n = \frac{7}{9} \left( \frac{10}{9} (10^n - 1) - n \right) \]

### Step 6: Simplify the expression
Now we can simplify:
\[ S_n = \frac{7}{81} (10^{n+1} - 10 - 9n) \]

### Final Answer
Thus, the sum of the first \(n\) terms of the series is:
\[ S_n = \frac{7}{81} (10^{n+1} - 10 - 9n) \]