`S_(oo)` of the following infinite G.P. `1,1/2,1/(2^(2))` To `oo` is ………………….

Answer

To find the sum of the infinite geometric progression (G.P.) given by the terms \(1, \frac{1}{2}, \frac{1}{2^2}, \ldots\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the First Term (a)**:
The first term \(a\) of the G.P. is the first term of the sequence.
\[ a = 1 \]

2. **Identify the Common Ratio (r)**:
The common ratio \(r\) can be found by dividing the second term by the first term.
\[ r = \frac{\text{second term}}{\text{first term}} = \frac{\frac{1}{2}}{1} = \frac{1}{2} \]

3. **Check the Condition for Convergence**:
For the sum of an infinite G.P. to exist, the absolute value of the common ratio must be less than 1.
\[ |r| = \left|\frac{1}{2}\right| < 1 \]
This condition is satisfied.

4. **Use the Formula for the Sum of an Infinite G.P.**:
The formula for the sum \(S_{\infty}\) of an infinite G.P. is given by:
\[ S_{\infty} = \frac{a}{1 - r} \]
Substituting the values of \(a\) and \(r\):
\[ S_{\infty} = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \]

5. **Final Result**:
Therefore, the sum of the infinite G.P. is:
\[ S_{\infty} = 2 \]