Question

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

• A. -2
• B. -4
• C. -12
• D. 8

Answer 1

The Correct Option is C

Given a geometric progression where the first term is denoted by \( a \), and the common ratio is \( r \), the first four terms are \( a, ar, ar^2, ar^3 \). According to the problem:

• \( a + ar = 12 \)
• \( ar^2 + ar^3 = 48 \)

The terms are alternately positive and negative. This implies \( r \) must be negative, such that even terms \( ar, ar^3 \) are negative. From the first equation, factor out \( a \):

\( a(1 + r) = 12 \) (Equation 1)

From the second equation, factor out \( ar^2 \):

\( ar^2(1 + r) = 48 \) (Equation 2)

Substitute \( ar^2 = \frac{ar \cdot a}{a} \) and rearrange:

From Equation 1, \( a = \frac{12}{1 + r} \)

Substitute \( a \) from Equation 1 into Equation 2:

\( \frac{12r^2}{1 + r}(1 + r) = 48 \)

\( 12r^2 = 48 \)

Divide by 12:

\( r^2 = 4 \)

\( r = -2 \) (since \( r \) is negative for alternately positive and negative terms)

Substitute \( r = -2 \) back into \( a + ar = 12 \):

\( a + a(-2) = 12 \)

\( a(1 - 2) = 12 \)

\( -a = 12 \)

\( a = -12 \)

Thus, the first term is \( -12 \).