If an infinite G.P. has the first term a and the sum 5, then which one of the following is correct?

Options

A. A. `alt-10`
B. B. `-10ltalt0`
C. C. `0ltalt10`
D. D. `agt10`

Answer

To solve the problem, we need to analyze the properties of an infinite geometric progression (G.P.) with the first term \( a \) and a sum of 5.

### Step-by-Step Solution:

1. **Understanding the Sum of an Infinite G.P.**:
The sum \( S \) of an infinite geometric progression is given by the formula:
\[ S = \frac{a}{1 - r} \]
where \( a \) is the first term and \( r \) is the common ratio.

2. **Setting Up the Equation**:
We are given that the sum \( S = 5 \). Therefore, we can set up the equation:
\[ \frac{a}{1 - r} = 5 \]

3. **Rearranging the Equation**:
To find a relationship between \( a \) and \( r \), we can rearrange the equation:
\[ a = 5(1 - r) \]
This simplifies to:
\[ a = 5 - 5r \]

4. **Considering the Condition for Convergence**:
For the sum of an infinite G.P. to converge, the common ratio \( r \) must satisfy:
\[ |r| < 1 \]
This means that \( r \) must be in the interval \( (-1, 1) \).

5. **Finding the Range of \( a \)**:
We can express \( r \) in terms of \( a \):
\[ r = \frac{5 - a}{5} \]
Since \( |r| < 1 \), we can write:
\[ -1 < \frac{5 - a}{5} < 1 \]

6. **Solving the Inequalities**:
- **Left Inequality**:
\[ -1 < \frac{5 - a}{5} \]
Multiplying both sides by 5:
\[ -5 < 5 - a \]
Rearranging gives:
\[ a < 10 \]

- **Right Inequality**:
\[ \frac{5 - a}{5} < 1 \]
Multiplying both sides by 5:
\[ 5 - a < 5 \]
Rearranging gives:
\[ a > 0 \]

7. **Conclusion**:
Combining both inequalities, we find:
\[ 0 < a < 10 \]
Therefore, the correct option is that \( a \) is between 0 and 10.

### Final Answer:
The correct option is **C: \( a \) is between 0 and 10**.