If `r ge1`, then the sum of infinite G.P. tends to (i) 0 (ii) `oo` (iii) 1 (iv) none of these

Options

A. 0
B. `oo`
C. 1
D. none of these

Answer

To solve the problem, we need to analyze the sum of an infinite geometric progression (G.P.) given that the common ratio \( r \) is greater than or equal to 1.

### Step-by-Step Solution:

1. **Understanding the Formula for the Sum of an Infinite G.P.:**
The sum of an infinite geometric progression is given by:
\[ S_{\infty} = \frac{a}{1 - r} \quad \text{if } |r| < 1 \]
and
\[ S_{\infty} = \frac{a}{r - 1} \quad \text{if } r > 1 \]
where \( a \) is the first term of the G.P.

2. **Case When \( r = 1 \):**
If \( r = 1 \), the formula becomes:
\[ S_{\infty} = \frac{a}{1 - 1} = \frac{a}{0} \]
This expression is undefined and tends towards infinity.

3. **Case When \( r > 1 \):**
If \( r > 1 \), we use the second formula:
\[ S_{\infty} = \frac{a}{r - 1} \]
Since \( r - 1 \) is positive (as \( r > 1 \)), the sum \( S_{\infty} \) will also be positive. However, as \( r \) increases, \( S_{\infty} \) tends to infinity.

4. **Conclusion:**
Therefore, for \( r \geq 1 \), the sum of the infinite G.P. tends to infinity.

5. **Final Answer:**
The correct option is (ii) \( \infty \).