Find the sum of first n terms of the series 0.7 + 0.77 + 0.777 + …..

Answer

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

### Step 1: Express the terms in a simplified form
The terms of the series can be expressed as:
- \( 0.7 = \frac{7}{10} \)
- \( 0.77 = \frac{77}{100} \)
- \( 0.777 = \frac{777}{1000} \)

We can generalize the \( n \)-th term of the series as:
\[ T_n = \frac{7 \times (10^n - 1)}{9 \times 10^n} \]
This is because \( 0.7 \) can be seen as \( 7/10 \), \( 0.77 \) as \( 77/100 \), and so on.

### Step 2: Rewrite the series
The sum of the first \( n \) terms can be written as:
\[ S_n = \frac{7}{10} + \frac{77}{100} + \frac{777}{1000} + \ldots \]
Factoring out \( 7 \):
\[ S_n = 7 \left( \frac{1}{10} + \frac{11}{100} + \frac{111}{1000} + \ldots \right) \]

### Step 3: Identify the pattern in the series
Notice that:
\[ \frac{1}{10} = 0.1, \quad \frac{11}{100} = 0.11, \quad \frac{111}{1000} = 0.111 \]
This can be expressed as:
\[ S_n = 7 \left( 0.1 + 0.11 + 0.111 + \ldots \right) \]

### Step 4: Convert to a geometric series
We can express \( 0.1, 0.11, 0.111 \) in terms of a geometric series:
\[ 0.1 + 0.11 + 0.111 + \ldots = 0.1 \left( 1 + 0.1 + 0.01 + \ldots \right) \]
The series inside the parentheses is a geometric series with first term \( 1 \) and common ratio \( 0.1 \).

### Step 5: Sum the geometric series
The sum of the first \( n \) terms of a geometric series is given by:
\[ S = \frac{a(1 - r^n)}{1 - r} \]
where \( a = 1 \) and \( r = 0.1 \):
\[ S_n = \frac{1(1 - (0.1)^n)}{1 - 0.1} = \frac{1 - (0.1)^n}{0.9} \]

### Step 6: Substitute back into the sum
Now substituting back into our expression for \( S_n \):
\[ S_n = 7 \left( 0.1 \cdot \frac{1 - (0.1)^n}{0.9} \right) \]
This simplifies to:
\[ S_n = \frac{7}{9} (1 - (0.1)^n) \]

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