The sum of 10 terms of the series `0.7+0.77+0.777+………`is -

Answer

To find the sum of the first 10 terms of the series \(0.7 + 0.77 + 0.777 + \ldots\), we can follow these steps:

### Step 1: Identify the pattern in the series
The series can be expressed as:
- First term: \(0.7 = \frac{7}{10}\)
- Second term: \(0.77 = \frac{77}{100} = \frac{7 \times 11}{100}\)
- Third term: \(0.777 = \frac{777}{1000} = \frac{7 \times 111}{1000}\)

We can see that each term can be represented as:
\[ \text{n-th term} = \frac{7 \times (10^n - 1)/9}{10^n} \]
This can be simplified to:
\[ \text{n-th term} = \frac{7}{9} \left(1 - \frac{1}{10^n}\right) \]

### Step 2: Write the sum of the first 10 terms
The sum of the first 10 terms can be expressed as:
\[ S_{10} = \sum_{n=1}^{10} \frac{7}{9} \left(1 - \frac{1}{10^n}\right) \]
This can be separated into two sums:
\[ S_{10} = \frac{7}{9} \left(\sum_{n=1}^{10} 1 - \sum_{n=1}^{10} \frac{1}{10^n}\right) \]

### Step 3: Calculate the first sum
The first sum is simply:
\[ \sum_{n=1}^{10} 1 = 10 \]

### Step 4: Calculate the second sum
The second sum is a geometric series:
\[ \sum_{n=1}^{10} \frac{1}{10^n} = \frac{\frac{1}{10}(1 - (\frac{1}{10})^{10})}{1 - \frac{1}{10}} = \frac{1}{10} \cdot \frac{1 - \frac{1}{10^{10}}}{\frac{9}{10}} = \frac{1 - \frac{1}{10^{10}}}{9} \]

### Step 5: Substitute back into the sum
Now substituting back into the equation for \(S_{10}\):
\[ S_{10} = \frac{7}{9} \left(10 - \frac{1 - \frac{1}{10^{10}}}{9}\right) \]

### Step 6: Simplify the expression
Simplifying this gives:
\[ S_{10} = \frac{7}{9} \left(10 - \frac{1}{9} + \frac{1}{9 \cdot 10^{10}}\right) \]
\[ = \frac{7}{9} \left(\frac{90 - 1}{9} + \frac{1}{9 \cdot 10^{10}}\right) \]
\[ = \frac{7}{9} \cdot \frac{89}{9} + \frac{7}{9 \cdot 10^{10}} \]
\[ = \frac{7 \cdot 89}{81} + \frac{7}{9 \cdot 10^{10}} \]

### Final Result
Thus, the sum of the first 10 terms of the series is:
\[ S_{10} = \frac{623}{81} + \frac{7}{9 \cdot 10^{10}} \]