Find the sum to infinity of the G.P; \( 2,1, \frac{1}{2}, \frac{1}{4}, \cdots \)

To find the sum to infinity of the given geometric progression (G.P.): \[ 2, \; 1, \; \frac{1}{2}, \; \frac{1}{4}, \; \cdots \] **Step 1: Identify the First Term and Common Ratio** - **First term (\(a\))**: The first term of the G.P. is \( a = 2 \). - **Common ratio (\(r\))**: To find the common ratio, divide the second term by the first term: \[ r = \frac{1}{2} = \frac{1}{2} \] **Step 2: Check if the Series Converges** For the sum to infinity of a G.P. to exist (i.e., to converge), the common ratio must satisfy: \[ |r| < 1 \] In this case, \( r = \frac{1}{2} \), and: \[ \left| \frac{1}{2} \right| = \frac{1}{2} < 1 \] Since the condition is met, the series converges. **Step 3: Use the Formula for the Sum to Infinity** The sum to infinity (\( S_\infty \)) of a G.P. is given by: \[ S_\infty = \frac{a}{1 - r} \] Substituting the known values: \[ S_\infty = \frac{2}{1 - \frac{1}{2}} = \frac{2}{\frac{1}{2}} = 4 \] **Conclusion** The sum to infinity of the given geometric progression is: \[ \boxed{4} \]