Question

Which choice is the explicit formula for the following geometric sequence?

Answer 1

$-0.1=-\dfrac15\times0.5$
$0.02=-\dfrac15\times(-0.1)$
$-0.004=-\dfrac15\times0.02$
$0.0008=-\dfrac15\times(-0.004)$

This means the common ratio between the terms in the sequence is $-\dfrac15$, so you know the recursive formula is

$a_n=-\dfrac15a_{n-1}$

which you can solve recursively to find an explicit formula in terms of $a_1=0.5$.

$a_n=-\dfrac15a_{n-1}=\left(-\dfrac15\right)^2a_{n-2}=\left(-\dfrac15\right)^3a_{n-3}=\cdots=\left(-\dfrac15\right)^{n-1}a_1$

So the explicit formula is

$a_n=\dfrac12\left(-\dfrac15\right)^{n-1}=0.5\times(-0.2)^{n-1}$

Answer 2

Geometric sequence
an=a1(r)^(n-1)
r=common ratio
a1=first term
an=nth term

the common ratio is found by dividing a term by the term before it

-0.1/0.5=-1/5

the common ratio is -1/5
the 1th term is 0.05

the formula is
an=0.5(-1/5)^(n-1) or
an=-2.5(-1/5)^n