8 + 88 + 888 +…n terms =

Options

A. `(80)/(81)(10^(n) - 1) - (8n)/(9)`
B. `(10)/(81) (10^(n) - 1)`
C. `(80)/(81) (10^(n) - 1) + (8n)/(9)`
D. none of these

Answer

To find the sum of the series \( S = 8 + 88 + 888 + \ldots \) for \( n \) terms, we can follow these steps:

### Step 1: Identify the pattern
The terms in the series can be represented as:
- First term: \( 8 = 8 \times 1 \)
- Second term: \( 88 = 8 \times 11 \)
- Third term: \( 888 = 8 \times 111 \)

We can see that each term can be expressed as \( 8 \times (10^k - 1)/9 \) for \( k = 1, 2, 3, \ldots, n \).

### Step 2: Express the \( n \)-th term
The \( n \)-th term can be expressed as:
\[ T_n = 8 \times \frac{10^n - 1}{9} \]

### Step 3: Write the sum of the series
The sum of the first \( n \) terms can be written as:
\[ S_n = 8 + 88 + 888 + \ldots + T_n \]

### Step 4: Rewrite the sum
We can rewrite the sum using the formula for the \( n \)-th term:
\[ S_n = 8 \left( 1 + 11 + 111 + \ldots + \frac{10^n - 1}{9} \right) \]

### Step 5: Factor out the common term
Factoring out \( 8 \):
\[ S_n = 8 \left( \frac{1}{9} \left( 10^n - 1 \right) \right) \]

### Step 6: Calculate the sum
Now, we can simplify:
\[ S_n = \frac{8}{9} \left( 10^n - 1 \right) \]

### Step 7: Final expression
Thus, the sum of the series \( S_n \) for \( n \) terms is:
\[ S_n = \frac{8}{9} (10^n - 1) \]