The common ratio of a GP is `-(4)/(5)` and the sum to infinity is `(80)/(9)` . The first term of GP is

Options

A. 16
B. 8
C. 12
D. None of these

Answer

To find the first term of the geometric progression (GP) given the common ratio and the sum to infinity, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Common ratio \( r = -\frac{4}{5} \)
- Sum to infinity \( S = \frac{80}{9} \)

2. **Use the Formula for the Sum of a GP**:
The formula for the sum to infinity of a GP is given by:
\[ S = \frac{a}{1 - r} \]
where \( a \) is the first term and \( r \) is the common ratio.

3. **Substitute the Known Values into the Formula**:
We can substitute the values of \( S \) and \( r \) into the formula:
\[ \frac{80}{9} = \frac{a}{1 - \left(-\frac{4}{5}\right)} \]

4. **Simplify the Denominator**:
The denominator becomes:
\[ 1 - \left(-\frac{4}{5}\right) = 1 + \frac{4}{5} = \frac{5}{5} + \frac{4}{5} = \frac{9}{5} \]
Thus, the equation now looks like:
\[ \frac{80}{9} = \frac{a}{\frac{9}{5}} \]

5. **Cross-Multiply to Solve for \( a \)**:
Cross-multiplying gives:
\[ 80 \cdot \frac{9}{5} = 9a \]
Simplifying the left side:
\[ \frac{720}{5} = 9a \]
\[ 144 = 9a \]

6. **Isolate \( a \)**:
Divide both sides by 9:
\[ a = \frac{144}{9} = 16 \]

### Conclusion:
The first term \( a \) of the GP is \( 16 \).