The sum of first two terms of a G.P. is `5/3` and the sum to infinity of the series is 3. Find the first term :

Options

A. 1
B. `2/3`
C. 5
D. both (a) and (c)

Answer

To solve the problem, we need to find the first term \( A \) of a geometric progression (G.P.) given the following conditions:

1. The sum of the first two terms of the G.P. is \( \frac{5}{3} \).
2. The sum to infinity of the series is \( 3 \).

### Step 1: Set up the equations

The first term of the G.P. is \( A \) and the common ratio is \( R \). The first two terms of the G.P. are \( A \) and \( AR \).

From the first condition, we can write the equation:
\[ A + AR = \frac{5}{3} \]
Factoring out \( A \):
\[ A(1 + R) = \frac{5}{3} \quad \text{(Equation 1)} \]

The sum to infinity of a G.P. is given by the formula:
\[ S_{\infty} = \frac{A}{1 - R} \]
From the second condition, we can write:
\[ \frac{A}{1 - R} = 3 \quad \text{(Equation 2)} \]

### Step 2: Express \( A \) in terms of \( R \)

From Equation 1, we can express \( A \):
\[ A = \frac{5}{3(1 + R)} \]

### Step 3: Substitute \( A \) in Equation 2

Now, substitute \( A \) from Equation 1 into Equation 2:
\[ \frac{\frac{5}{3(1 + R)}}{1 - R} = 3 \]

### Step 4: Simplify the equation

Cross-multiplying gives:
\[ 5 = 3(1 - R) \cdot 3(1 + R) \]
This simplifies to:
\[ 5 = 9(1 - R^2) \]

### Step 5: Rearranging the equation

Rearranging gives:
\[ 9(1 - R^2) = 5 \]
\[ 9 - 9R^2 = 5 \]
\[ 9R^2 = 4 \]
\[ R^2 = \frac{4}{9} \]
Taking the square root gives:
\[ R = \frac{2}{3} \quad \text{or} \quad R = -\frac{2}{3} \]

### Step 6: Find \( A \) for both values of \( R \)

**Case 1:** If \( R = \frac{2}{3} \):
Substituting back into Equation 1:
\[ A(1 + \frac{2}{3}) = \frac{5}{3} \]
\[ A \cdot \frac{5}{3} = \frac{5}{3} \]
Thus,
\[ A = 1 \]

**Case 2:** If \( R = -\frac{2}{3} \):
Substituting back into Equation 1:
\[ A(1 - \frac{2}{3}) = \frac{5}{3} \]
\[ A \cdot \frac{1}{3} = \frac{5}{3} \]
Thus,
\[ A = 5 \]

### Conclusion

The first term \( A \) can be either \( 1 \) or \( 5 \).

### Final Answer:
The first term \( A \) is \( 1 \) or \( 5 \).

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