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: Set up the GP.
Let the first term be \(a\) and the common ratio be \(r\). The first two terms give \(a + ar = a(1+r) = 12\). Call this equation (1).

Step 2: Write the second condition.
The third and fourth terms give \(ar^2 + ar^3 = ar^2(1+r) = 48\). Call this equation (2).

Step 3: Divide to find r.
Dividing (2) by (1) cancels the \((1+r)\) factor and gives \(\frac{ar^2(1+r)}{a(1+r)} = \frac{48}{12}\), so \(r^2 = 4\), which means \(r = 2\) or \(r = -2\).

Step 4: Use the sign condition.
The problem says the terms alternate between positive and negative. A positive ratio keeps every term the same sign as \(a\), so \(r = 2\) is rejected. Only \(r = -2\) makes the terms flip sign each time.

Step 5: Solve for a.
Put \(r = -2\) into (1): \(a(1 + (-2)) = 12\), so \(a(-1) = 12\), which gives \(a = -12\).

Step 6: Check the other options.
Option A (\(-2\)) and option B (\(-4\)) do not satisfy \(a(1+r)=12\) once \(r=-2\) is fixed, and option D (\(8\)) gives a positive first term, which contradicts the alternating pattern once \(r\) is negative.

Final Answer:
The first term is \(-12\). \[ \boxed{a = -12} \]