Find the sum to n terms of the sequence 8,88,888,8888,……

Explanation

To find the sum to n terms of the sequence 8, 88, 888, 8888, ..., we can express the terms in a more manageable form.

### Step-by-Step Solution:

1. **Identify the Sequence**:
The sequence can be written as:
- 1st term: \( 8 \)
- 2nd term: \( 88 = 8 \times 11 \)
- 3rd term: \( 888 = 8 \times 111 \)
- 4th term: \( 8888 = 8 \times 1111 \)

We can express the nth term as:
\[ a_n = 8 \times (10^n - 1) / 9 \]

2. **Sum of the Sequence**:
To find the sum of the first n terms, we write:
\[ S_n = 8 + 88 + 888 + \ldots + a_n \]

This can be rewritten as:
\[ S_n = 8 \left( 1 + 11 + 111 + \ldots + (10^n - 1)/9 \right) \]

3. **Simplifying the Sum**:
Notice that \( 11 = 10 + 1 \), \( 111 = 100 + 10 + 1 \), and so forth. We can express this in terms of a geometric series:
\[ S_n = 8 \left( \frac{10^n - 1}{9} \right) \]

4. **Using the Formula for the Sum of a Geometric Series**:
The sum of the series can be calculated using the formula for the sum of a geometric series:
\[ S_n = 8 \left( \frac{10(10^n - 1)}{9} \right) \]

5. **Final Expression**:
Thus, the sum of the first n terms can be expressed as:
\[ S_n = \frac{8}{9} \left( 10^{n+1} - 10 \right) \]

### Final Answer:
The sum to n terms of the sequence is:
\[ S_n = \frac{8}{9} \left( 10^{n+1} - 10 \right) \]