Find the sum of the series :
(i) `8+88+888+` ... to n terms
(ii) `3+33+333+` ... to n terms
(iii) `0.7+0.77+0.777+`... to n terms

Explanation

To find the sum of the series for each part, we will break down the problem step by step.

### Part (i): Sum of the series \(8 + 88 + 888 + \ldots\) to \(n\) terms

1. **Factor out 8**:
\[ S_n = 8 + 88 + 888 + \ldots = 8(1 + 11 + 111 + \ldots) \]

2. **Rewrite the series**:
\[ S_n = 8(1 + 11 + 111 + \ldots) = 8 \left( \frac{9}{9} \right)(1 + 11 + 111 + \ldots) = \frac{8}{9}(9 + 99 + 999 + \ldots) \]

3. **Express numbers in terms of powers of 10**:
\[ 9 = 10 - 1, \quad 99 = 100 - 1, \quad 999 = 1000 - 1 \]
Thus,
\[ S_n = \frac{8}{9} \left( (10 + 100 + 1000 + \ldots) - n \right) \]

4. **Identify the geometric series**:
The series \(10 + 100 + 1000 + \ldots\) is a geometric series with first term \(10\) and common ratio \(10\):
\[ S = 10 \frac{10^n - 1}{10 - 1} = \frac{10}{9}(10^n - 1) \]

5. **Combine results**:
\[ S_n = \frac{8}{9} \left( \frac{10}{9}(10^n - 1) - n \right) \]
\[ S_n = \frac{80}{81}(10^n - 1) - \frac{8n}{9} \]

### Part (ii): Sum of the series \(3 + 33 + 333 + \ldots\) to \(n\) terms

1. **Factor out 3**:
\[ S_n = 3 + 33 + 333 + \ldots = 3(1 + 11 + 111 + \ldots) \]

2. **Rewrite the series**:
\[ S_n = 3 \left( \frac{9}{9} \right)(1 + 11 + 111 + \ldots) = \frac{1}{3}(9 + 99 + 999 + \ldots) \]

3. **Express numbers in terms of powers of 10**:
\[ S_n = \frac{1}{3} \left( (10 + 100 + 1000 + \ldots) - n \right) \]

4. **Identify the geometric series**:
\[ S = 10 \frac{10^n - 1}{10 - 1} = \frac{10}{9}(10^n - 1) \]

5. **Combine results**:
\[ S_n = \frac{1}{3} \left( \frac{10}{9}(10^n - 1) - n \right) \]
\[ S_n = \frac{10}{27}(10^n - 1) - \frac{n}{3} \]

### Part (iii): Sum of the series \(0.7 + 0.77 + 0.777 + \ldots\) to \(n\) terms

1. **Factor out 0.7**:
\[ S_n = 0.7 + 0.77 + 0.777 + \ldots = 0.7(1 + 1.1 + 1.11 + \ldots) \]

2. **Rewrite the series**:
\[ S_n = 0.7 \left( \frac{9}{9} \right)(1 + 1.1 + 1.11 + \ldots) = \frac{7}{9}(0.9 + 0.99 + 0.999 + \ldots) \]

3. **Express numbers in terms of powers of 10**:
\[ S_n = \frac{7}{9} \left( (1 - 0.1) + (1 - 0.01) + (1 - 0.001) + \ldots \right) \]

4. **Separate the terms**:
\[ S_n = \frac{7}{9} \left( n - 0.1(1 + 0.1 + 0.01 + \ldots) \right) \]

5. **Identify the geometric series**:
\[ S = 0.1 \frac{1 - (0.1)^n}{1 - 0.1} = \frac{1}{0.9}(1 - 0.1^n) \]

6. **Combine results**:
\[ S_n = \frac{7}{9} n - \frac{7}{81}(1 - 0.1^n) \]

### Summary of Results:
1. For \(8 + 88 + 888 + \ldots\) to \(n\) terms:
\[ S_n = \frac{80}{81}(10^n - 1) - \frac{8n}{9} \]

2. For \(3 + 33 + 333 + \ldots\) to \(n\) terms:
\[ S_n = \frac{10}{27}(10^n - 1) - \frac{n}{3} \]

3. For \(0.7 + 0.77 + 0.777 + \ldots\) to \(n\) terms:
\[ S_n = \frac{7}{9} n - \frac{7}{81}(1 - 0.1^n) \]