Question

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

• A. \(-2\)
• B. \(-4\)
• C. \(-12\)
• D. \(8\)

Hint:

Divide the sum of the 3rd and 4th terms by the sum of the first two to isolate \(r^2\), then use the alternating sign clue to pick the right root.

Answer 1

The Correct Option is C

Step 1: Force the ratio negative from the start.
Since the terms alternate sign, write $r = -k$ with $k > 0$. The first two terms give $a(1 - k) = 12$, call this (1).

Step 2: Write the next pair of terms.
The third and fourth terms are $ar^2 + ar^3 = a k^2 (1 - k) = 48$, call this (2), since $r^2 = k^2$ and $r^3 = -k^3$.

Step 3: Cancel the common factor.
Dividing (2) by (1) removes $a(1-k)$ from both sides: $k^2 = \frac{48}{12} = 4$, so $k = 2$ (rejecting the negative root since $k > 0$).

Step 4: Recover r and a.
So $r = -k = -2$. Substitute back into (1): $a(1 - 2) = 12$, giving $-a = 12$, so $a = -12$.

Step 5: Verify with actual terms.
The terms become $-12, 24, -48, 96$. Check: first two sum to $-12 + 24 = 12$, correct. Third and fourth sum to $-48 + 96 = 48$, correct, and the signs alternate as required.

Final Answer:
This confirms the first term. \[ \boxed{a = -12} \]