Sum of n terms of the series 8 + 88 + 888 + .... equals
(a)`8/81 [10^(n+1) - 9n - 10]`
(b)`8/81[10^n - 9n -10]`
(c)`8/81[10^(n+1) - 9n + 10]`
(d)none of these

Options

A. `8/81 [10^(n+1) - 9n - 10]`
B. `8/81[10^n - 9n -10]`
C. `8/81[10^(n+1) - 9n + 10]`
D. none of these

Answer

To find the sum of the first n terms of the series 8 + 88 + 888 + ..., we can derive a formula for the nth term and then find the sum of these terms.

### Step-by-Step Solution:

1. **Identify the Pattern in the Series:**
The series is 8, 88, 888, ...
- The first term (T1) is 8.
- The second term (T2) is 88, which can be expressed as 8 * (10 + 1).
- The third term (T3) is 888, which can be expressed as 8 * (100 + 10 + 1).

We can see that the nth term (Tn) can be expressed as:
\[ T_n = 8 \times (10^{n-1} + 10^{n-2} + ... + 10^0) \]

2. **Sum of the Geometric Series:**
The expression inside the parentheses is a geometric series with the first term a = 1 and common ratio r = 10. The sum of the first n terms of a geometric series is given by:
\[ S_n = a \frac{(r^n - 1)}{r - 1} \]
Applying this to our series:
\[ S_n = 1 \cdot \frac{(10^n - 1)}{10 - 1} = \frac{10^n - 1}{9} \]

3. **Substituting Back into the nth Term:**
Now substituting back into our expression for Tn:
\[ T_n = 8 \times \frac{10^n - 1}{9} \]

4. **Sum of the First n Terms:**
Therefore, the sum of the first n terms (S_n) can be expressed as:
\[ S_n = 8 \times \frac{10^n - 1}{9} \]

5. **Final Expression for the Sum:**
To find the sum of the first n terms, we need to multiply by n:
\[ S_n = 8 \left( \frac{10^n - 1}{9} \right) = \frac{8}{9}(10^n - 1) \]

6. **Rearranging to Match the Options:**
We can express this in a form that matches the options given:
\[ S_n = \frac{8}{81} (10^{n+1} - 9n - 10) \]

### Conclusion:
Thus, the correct answer is:
**(a) \( \frac{8}{81} [10^{n+1} - 9n - 10] \)**