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To find the first and the fifth terms of a geometric sequence, we need to write the equation for the $nth$ term of the sequence.
$a_{n}=a_{1}⋅r_{n−1} $
We are given the $third$ and $fourth$ terms of the sequence, $-12$ and $24.$ Their ratio gives us the value of $r.$
$a_{3}a_{4} =-1224 ⇒r=-2 $
We can find the first term by substituting $r=-2$ and $a_{3}=-12$ in the equation for the $n_{th}$ term of the sequence.
The first term of the sequence is $-3.$ Therefore, the equation for the $n_{th}$ term of the sequence can be written as follows.
$a_{n}=a_{1}⋅r_{n−1}⇒a_{n}=-3⋅(-2)_{n−1} $
Substituting $5$ for $n$ gives us the fifth term of the sequence.
The fifth term of the sequence is $-48.$

$a_{n}=a_{1}⋅r_{n−1}$

SubstituteIIISubstitute $r=-2,n=3,a_{3}=-12$

$-12=a_{1}⋅(-2)_{3−1}$

Solve for $a_{1}$

SubTermSubtract term

$-12=a_{1}⋅(-2)_{2}$

CalcPowCalculate power

$-12=a_{1}⋅4$

DivEqn$LHS/4=RHS/4$

$-3=a_{1}$

RearrangeEqnRearrange equation

$a_{1}=-3$