Find the sum of n terms of the following expression.
8 + 88 + 888 + ........ ?

Options

A. `(8(10^n-9^n))/81`
B. `(8(10^(n+1)-10-9n))/81`
C. `8(10^(n-1)-10)`
D. `8(10^(n+1)-10)`

Answer

To find the sum of the first n terms of the sequence 8, 88, 888, ..., we can follow these steps:

### Step 1: Identify the Pattern
The terms in the sequence can be expressed in a more manageable form. The first term is 8, the second term is 88, the third term is 888, and so on. Each term can be represented as:
- 1st term: 8 = 8
- 2nd term: 88 = 8 × 11
- 3rd term: 888 = 8 × 111
- ...
- n-th term: 8 × (10^n - 1) / 9 (This is derived from the formula for the sum of a geometric series).

### Step 2: Write the n-th Term
The n-th term can be generalized as:
\[ T_n = 8 \times \frac{10^n - 1}{9} \]

### Step 3: Find the Sum of n Terms
To find the sum of the first n terms (S_n), we can sum up the first n terms:
\[ S_n = T_1 + T_2 + T_3 + ... + T_n \]
\[ S_n = 8 \left( \frac{10^1 - 1}{9} + \frac{10^2 - 1}{9} + \frac{10^3 - 1}{9} + ... + \frac{10^n - 1}{9} \right) \]

### Step 4: Factor Out the Common Terms
We can factor out \( \frac{8}{9} \):
\[ S_n = \frac{8}{9} \left( (10^1 - 1) + (10^2 - 1) + (10^3 - 1) + ... + (10^n - 1) \right) \]
\[ S_n = \frac{8}{9} \left( (10^1 + 10^2 + 10^3 + ... + 10^n) - n \right) \]

### Step 5: Use the Formula for the Sum of a Geometric Series
The sum of the geometric series \( 10^1 + 10^2 + ... + 10^n \) can be calculated using the formula:
\[ S = a \frac{(r^n - 1)}{(r - 1)} \]
where \( a = 10 \), \( r = 10 \), and \( n \) is the number of terms.
Thus,
\[ S = 10 \frac{(10^n - 1)}{(10 - 1)} = \frac{10}{9} (10^n - 1) \]

### Step 6: Substitute Back into the Sum
Now we substitute this back into our expression for \( S_n \):
\[ S_n = \frac{8}{9} \left( \frac{10}{9} (10^n - 1) - n \right) \]
\[ S_n = \frac{8}{81} (10^{n+1} - 10 - 9n) \]

### Final Result
Thus, the sum of the first n terms of the sequence is:
\[ S_n = \frac{8}{81} (10^{n+1} - 10 - 9n) \]