The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is

Options

A. `-4`
B. `-12`
C. 12
D. 4

Answer

To solve the problem step by step, let's denote the first term of the geometric progression (GP) as \( a \) and the common ratio as \( r \).

### Step 1: Set up the equations based on the problem statement.

From the problem, we know:
1. The sum of the first two terms is 12:
\[ a + ar = 12 \quad \text{(Equation 1)} \]
2. The sum of the third and fourth terms is 48:
\[ ar^2 + ar^3 = 48 \quad \text{(Equation 2)} \]

### Step 2: Simplify Equation 2.

We can factor out \( ar^2 \) from Equation 2:
\[ ar^2(1 + r) = 48 \]

### Step 3: Express \( ar^2 \) in terms of \( a \) and \( r \).

From Equation 1, we can express \( ar \) as:
\[ ar = 12 - a \]
Now substituting \( ar \) into Equation 2:
\[ ar^2 = \frac{(12 - a)}{r} \]
Now substituting this into the modified Equation 2:
\[ \frac{(12 - a)r^2}{r}(1 + r) = 48 \]
This simplifies to:
\[ (12 - a)r(1 + r) = 48 \]

### Step 4: Substitute \( a \) from Equation 1 into the equation.

From Equation 1, we can express \( a \) as:
\[ a = 12 - ar \]
Substituting this into the equation:
\[ (12 - (12 - ar))r(1 + r) = 48 \]
This simplifies to:
\[ ar(1 + r) = 48 \]

### Step 5: Substitute \( ar \) from Equation 1.

Using \( ar = 12 - a \):
\[ (12 - a)(1 + r) = 48 \]

### Step 6: Solve for \( r \).

Now we have two equations:
1. \( a + ar = 12 \)
2. \( (12 - a)(1 + r) = 48 \)

From the first equation, we can express \( ar \) as:
\[ ar = 12 - a \]
Substituting this into the second equation gives:
\[ (12 - a)(1 + r) = 48 \]

### Step 7: Solve the equations simultaneously.

We can solve for \( r \) using the relationship derived from the equations.

1. From \( ar = 12 - a \), we can substitute into the second equation:
\[ ar^2 + ar^3 = 48 \]
becomes
\[ (12 - a)r^2 + (12 - a)r^3 = 48 \]

### Step 8: Find the value of \( r \).

We can simplify and solve for \( r \):
\[ (12 - a)r^2(1 + r) = 48 \]
Using \( a + ar = 12 \) and substituting \( ar \):
\[ ar^2 + ar^3 = 48 \]

### Step 9: Determine the sign of \( r \).

Since the terms are alternately positive and negative, we find that \( r \) must be negative.

### Step 10: Solve for \( a \).

Finally, we find \( a \):
\[ a + ar = 12 \]
Substituting \( r = -2 \):
\[ a - 2a = 12 \]
This simplifies to:
\[ -a = 12 \implies a = -12 \]

### Conclusion

Thus, the first term of the geometric progression is:
\[ \boxed{-12} \]