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To find the first and the fourth terms of a geometric sequence, we need to write the equation for the $n_{th}$ term of the sequence.
$a_{n}=a_{1}⋅r_{n−1} $
We are given the $second$ and $third$ terms of the sequence, $3$ and $1.$ Their ratio gives us the value of $r.$
$a_{2}a_{3} =31 ⇒r=31 $
We can find the first term by substituting $r=31 $ and $a_{2}=3$ in the equation.
The first term of the sequence is $9.$ 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}=9⋅(31 )_{n−1} $
Substituting $4$ for $n$ gives us the fourth term of the sequence.
The fourth term of the sequence is $31 .$

$a_{n}=a_{1}⋅r_{n−1}$

SubstituteIIISubstitute $r=31 ,n=2,a_{2}=3$

$3=a_{1}⋅(31 )_{2−1}$

Solve for $a_{1}$

SubTermSubtract term

$3=a_{1}⋅(31 )_{1}$

ExponentOne$a_{1}=a$

$3=a_{1}⋅31 $

MultEqn$LHS⋅3=RHS⋅3$

$9=a_{1}$

RearrangeEqnRearrange equation

$a_{1}=9$

$a_{n}=9(31 )_{n−1}$

Substitute$n=4$

$a_{4}=9(31 )_{4−1}$

Simplify right-hand side

SubTermSubtract term

$a_{4}=9(31 )_{3}$

CalcPowCalculate power

$a_{4}=9(271 )$

MoveLeftFacToNumOne$a⋅b1 =ba $

$a_{4}=279 $

ReduceFrac$ba =b/9a/9 $

$a_{4}=31 $