Find the sum of 50 terms of the sequence:
7,7.7,7.77,7.777……..,

Answer

To find the sum of the first 50 terms of the sequence: 7, 7.7, 7.77, 7.777, ..., we can follow these steps:

### Step 1: Identify the pattern in the sequence
The sequence can be expressed in a more manageable form. Each term can be represented as:
- First term: \( 7 = 7.0 \)
- Second term: \( 7.7 = 7 + 0.7 \)
- Third term: \( 7.77 = 7 + 0.77 \)
- Fourth term: \( 7.777 = 7 + 0.777 \)

We can see that each term can be written as:
\[ a_n = 7 + \frac{7}{10} + \frac{7}{100} + \ldots + \frac{7}{10^n} \]

### Step 2: Rewrite the terms
We can express the sequence in a more general form:
\[ a_n = 7 + 7 \left( \frac{1}{10} + \frac{1}{100} + \ldots + \frac{1}{10^n} \right) \]

### Step 3: Identify the geometric series
The series \( \frac{1}{10} + \frac{1}{100} + \ldots + \frac{1}{10^n} \) is a geometric series where:
- First term \( A = \frac{1}{10} \)
- Common ratio \( r = \frac{1}{10} \)
- Number of terms \( n \)

The sum of the first \( n \) terms of a geometric series is given by:
\[ S_n = A \frac{1 - r^n}{1 - r} \]

### Step 4: Calculate the sum of the geometric series
Substituting the values into the formula:
\[ S_n = \frac{1}{10} \cdot \frac{1 - \left(\frac{1}{10}\right)^n}{1 - \frac{1}{10}} = \frac{1}{10} \cdot \frac{1 - \frac{1}{10^n}}{\frac{9}{10}} = \frac{1 - \frac{1}{10^n}}{9} \]

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

### Step 6: Calculate the sum of the first 50 terms
Now we need to find the sum of the first 50 terms:
\[ S_{50} = a_1 + a_2 + a_3 + \ldots + a_{50} \]

Using the expression for \( a_n \):
\[ S_{50} = \sum_{n=1}^{50} \left( 7 + \frac{7}{9} \left(1 - \frac{1}{10^n}\right) \right) \]
\[ S_{50} = 50 \cdot 7 + \frac{7}{9} \sum_{n=1}^{50} \left(1 - \frac{1}{10^n}\right) \]

### Step 7: Simplify the sum
The first part is straightforward:
\[ 50 \cdot 7 = 350 \]

Now for the second part:
\[ \sum_{n=1}^{50} 1 = 50 \]
\[ \sum_{n=1}^{50} \frac{1}{10^n} = \frac{\frac{1}{10}(1 - \left(\frac{1}{10}\right)^{50})}{1 - \frac{1}{10}} = \frac{1}{9} \left(1 - \frac{1}{10^{50}}\right) \]

### Step 8: Combine the results
Putting it all together:
\[ S_{50} = 350 + \frac{7}{9} \left(50 - \frac{1}{9} \left(1 - \frac{1}{10^{50}}\right)\right) \]

### Final Calculation
Calculating the final expression will give us the sum of the first 50 terms of the sequence.