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 terms of the geometric progression are alternately positive and negative, then the first term is

• A. $4$
• B. $-4$
• C. $-12$
• D. $12$

Answer 1

The correct option is C

$-12$

Explanation for correct option:
Find the given term of the geometric progression

Given: First two terms of geometric progression add up to $12$

that is, $a+ar=12....\left(I\right)$

Sum of the third and the fourth terms is $48$.

that is, $a{r}^2+a{r}^3=12....\left(II\right)$

Let us assume $a,ar,a{r}^2,a{r}^3$ be the $4$ terms of GP.

Using equation, $\left(I\right)$ and$\left(II\right)$

we get,

$$\begin{gathered} a(1+r)=12....\left(III\right) \\ a{r}^2(1+r)=48....\left(IV\right) \end{gathered}$$

Divide$\left(III\right)$by $\left(IV\right)$ we get,

$\frac{1}{{\mathrm{r}}^2}=\frac{1}{4}$

$r=2or-2$

Since alternate terms are negative, we take $r=-2$

Put $r$ in $\left(III\right)$, we get,

$$\begin{gathered} a(1-2)=12 \\ a=-12 \end{gathered}$$

Hence , the first term of the geometric progression is $-12$, so option (C) is the correct answer.