IF `A=(1/0.4) +(1/0.04)+1/(0.004)`+………up to 8 terms then what is the value of A

Options

A. 27272727.5
B. 25252525.5
C. 27777777.5
D. 25555555.5

Answer

To solve the problem, we need to find the value of \( A = \frac{1}{0.4} + \frac{1}{0.04} + \frac{1}{0.004} + \ldots \) up to 8 terms.

### Step-by-Step Solution:

1. **Identify the Pattern**:
The denominators of the terms are \( 0.4, 0.04, 0.004, \ldots \). We can express these in a more manageable form:
\[ 0.4 = \frac{4}{10} = \frac{10}{4}, \quad 0.04 = \frac{4}{100} = \frac{100}{4}, \quad 0.004 = \frac{4}{1000} = \frac{1000}{4} \]
Thus, we can rewrite the series as:
\[ A = \frac{10}{4} + \frac{100}{4} + \frac{1000}{4} + \ldots \]

2. **Factor Out the Common Denominator**:
We can factor out \( \frac{1}{4} \) from the series:
\[ A = \frac{1}{4} \left( 10 + 100 + 1000 + \ldots \right) \]

3. **Recognize the Series**:
The series inside the parentheses is a geometric series where the first term \( a = 10 \) and the common ratio \( r = 10 \). The number of terms \( n = 8 \).

4. **Sum of the Geometric Series**:
The sum \( S_n \) of the first \( n \) terms of a geometric series can be calculated using the formula:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substituting the values:
\[ S_8 = 10 \frac{10^8 - 1}{10 - 1} = 10 \frac{10^8 - 1}{9} \]

5. **Calculate \( S_8 \)**:
Calculate \( 10^8 \):
\[ 10^8 = 100000000 \]
Therefore,
\[ S_8 = 10 \frac{100000000 - 1}{9} = 10 \frac{99999999}{9} = \frac{999999990}{9} = 111111110 \]

6. **Final Calculation of \( A \)**:
Now substitute \( S_8 \) back into the expression for \( A \):
\[ A = \frac{1}{4} \times 111111110 = 27777777.5 \]

### Final Answer:
Thus, the value of \( A \) is \( 27777777.5 \).