If `S=1+a+a^2+a^3+a^4+……….to oo` then `a=`

Options

A. `(S)/(S-1)`
B. `(S)/(1-S)`
C. `(S-1)/(S)`
D. `(1-S)/(S)`

Explanation

To solve the problem, we need to find the value of \( a \) given the infinite series \( S = 1 + a + a^2 + a^3 + a^4 + \ldots \).

### Step-by-Step Solution:

1. **Identify the Series**: The series \( S = 1 + a + a^2 + a^3 + a^4 + \ldots \) is a geometric series where the first term \( a_1 = 1 \) and the common ratio \( r = a \).

2. **Formula for the Sum of an Infinite Geometric Series**: The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a_1}{1 - r} \]
where \( |r| < 1 \).

3. **Substituting the Values**: In our case, substituting the first term and the common ratio into the formula gives:
\[ S = \frac{1}{1 - a} \]

4. **Setting the Equation**: We know that \( S \) is equal to the expression we derived:
\[ S = \frac{1}{1 - a} \]

5. **Cross Multiplying**: To eliminate the fraction, we cross-multiply:
\[ S(1 - a) = 1 \]

6. **Expanding the Equation**: Expanding the left side gives:
\[ S - Sa = 1 \]

7. **Rearranging the Equation**: Rearranging the equation to isolate terms involving \( a \):
\[ Sa = S - 1 \]

8. **Solving for \( a \)**: Now, divide both sides by \( S \):
\[ a = \frac{S - 1}{S} \]

9. **Final Expression for \( a \)**: This simplifies to:
\[ a = 1 - \frac{1}{S} \]

### Conclusion:
Thus, the value of \( a \) is:
\[ a = 1 - \frac{1}{S} \]