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

Hint: We are given information about the first four terms of a geometric progression and we have to find the value of the first term. We know that a geometric progression, sometimes referred to as a geometric sequence in mathematics, is a series of non-zero numbers where each term following the first is obtained by multiplying the preceding one by a constant, non-zero value known as the common ratio. Using this definition, we will write the first four terms of the sequence and solve for the required value.

Complete step by step answer:
Let the first term of the geometric sequence be $a$ and its common difference be $r$ , then the first four terms of the geometric sequence will be $a,ar,a{r^2},a{r^3}$
We are given that the sum of first two terms is $12$ , so
$
  a + ar = 12 \\
   \Rightarrow a(1 + r) = 12 \\
 $
And we are also given that the sum of third and fourth term is $48$ , so
$
  a{r^2} + a{r^3} = 48 \\
   \Rightarrow a{r^2}(1 + r) = 48 \\
   \Rightarrow {r^2}(a(1 + r)) = 48 \\
 $
Now we put the value $a(1 + r) = 12$ in the above equation and get:
$
  12{r^2} = 48 \\
   \Rightarrow {r^2} = \dfrac{{48}}{{12}} \\
   \Rightarrow {r^2} = 4 \\
   \Rightarrow r = \pm 2 \\
 $
We are given that the alternate terms are negative, but if $r = 2$ then all the terms of the sequence will be either positive or negative, so we take $r = - 2$
We put this value of $r$ in the equation $a(1 - r) = 12$ and get:
$
  a(1 + ( - 2)) = 12 \\
   \Rightarrow a(1 - 2) = 12 \\
   \Rightarrow - a = 12 \\
   \Rightarrow a = - 12 \\
 $
The correct option is option C.

Note:
In geometric sequence each term is multiplied by the common difference, so if the common difference is positive, then multiplying it with a positive first term will yield all the terms as positive. But when the common difference is negative, a positive term will yield negative term and a negative term will yield positive term, thus we get an alternate series.