Sum of n terms:
`(i) 0.7 +0.77+0.777+`…………..
(ii) `0.6+0.66+0.666+`………….
(iii) `0.3+0.33+0.333+`…………..
(iv) `0.5+0.55+0.555+`………….

Answer

To find the sum of the first n terms for each of the given series, we can follow a systematic approach. Let's analyze each series step by step.

### (i) Series: \(0.7 + 0.77 + 0.777 + \ldots\)

1. **Identify the pattern**: The series can be rewritten as:
\[ S_n = 0.7 + 0.77 + 0.777 + \ldots \]
Each term can be expressed as:
\[ 0.7 = \frac{7}{10}, \quad 0.77 = \frac{77}{100}, \quad 0.777 = \frac{777}{1000}, \ldots \]
The nth term can be expressed as:
\[ a_n = \frac{7 \cdots 7}{10^n} \quad \text{(n times 7)} \]

2. **Rewrite the series**: Notice that:
\[ S_n = \frac{7}{10} + \frac{77}{100} + \frac{777}{1000} = \frac{7}{10} \left(1 + 0.1 + 0.01 + \ldots + 0.1^{n-1}\right) \]

3. **Use the formula for the sum of a geometric series**:
The sum of the first n terms of a geometric series is given by:
\[ S = a \frac{1 - r^n}{1 - r} \]
where \(a\) is the first term and \(r\) is the common ratio. Here, \(a = 1\) and \(r = 0.1\):
\[ S_n = \frac{7}{10} \cdot \frac{1 - (0.1)^n}{1 - 0.1} = \frac{7}{10} \cdot \frac{1 - (0.1)^n}{0.9} \]

4. **Simplify the expression**:
\[ S_n = \frac{7}{9} \left(1 - 0.1^n\right) \]

### (ii) Series: \(0.6 + 0.66 + 0.666 + \ldots\)

1. **Identify the pattern**: Similar to the first series:
\[ S_n = 0.6 + 0.66 + 0.666 + \ldots \]
Each term can be expressed as:
\[ 0.6 = \frac{6}{10}, \quad 0.66 = \frac{66}{100}, \quad 0.666 = \frac{666}{1000}, \ldots \]

2. **Rewrite the series**:
\[ S_n = \frac{6}{10} \left(1 + 0.1 + 0.01 + \ldots + 0.1^{n-1}\right) \]

3. **Use the sum formula**:
\[ S_n = \frac{6}{10} \cdot \frac{1 - (0.1)^n}{0.9} = \frac{6}{9} \left(1 - 0.1^n\right) \]

4. **Simplify**:
\[ S_n = \frac{2}{3} \left(1 - 0.1^n\right) \]

### (iii) Series: \(0.3 + 0.33 + 0.333 + \ldots\)

1. **Identify the pattern**:
\[ S_n = 0.3 + 0.33 + 0.333 + \ldots \]
Each term can be expressed as:
\[ 0.3 = \frac{3}{10}, \quad 0.33 = \frac{33}{100}, \quad 0.333 = \frac{333}{1000}, \ldots \]

2. **Rewrite the series**:
\[ S_n = \frac{3}{10} \left(1 + 0.1 + 0.01 + \ldots + 0.1^{n-1}\right) \]

3. **Use the sum formula**:
\[ S_n = \frac{3}{10} \cdot \frac{1 - (0.1)^n}{0.9} = \frac{3}{9} \left(1 - 0.1^n\right) \]

4. **Simplify**:
\[ S_n = \frac{1}{3} \left(1 - 0.1^n\right) \]

### (iv) Series: \(0.5 + 0.55 + 0.555 + \ldots\)

1. **Identify the pattern**:
\[ S_n = 0.5 + 0.55 + 0.555 + \ldots \]
Each term can be expressed as:
\[ 0.5 = \frac{5}{10}, \quad 0.55 = \frac{55}{100}, \quad 0.555 = \frac{555}{1000}, \ldots \]

2. **Rewrite the series**:
\[ S_n = \frac{5}{10} \left(1 + 0.1 + 0.01 + \ldots + 0.1^{n-1}\right) \]

3. **Use the sum formula**:
\[ S_n = \frac{5}{10} \cdot \frac{1 - (0.1)^n}{0.9} = \frac{5}{9} \left(1 - 0.1^n\right) \]

4. **Simplify**:
\[ S_n = \frac{5}{9} \left(1 - 0.1^n\right) \]

### Final Results:
1. \( S_n = \frac{7}{9} \left(1 - 0.1^n\right) \)
2. \( S_n = \frac{2}{3} \left(1 - 0.1^n\right) \)
3. \( S_n = \frac{1}{3} \left(1 - 0.1^n\right) \)
4. \( S_n = \frac{5}{9} \left(1 - 0.1^n\right) \)